Formulation of electro-chemico-osmotic processes in soils

Electrokinetic techniques have been used for various purposes including consolidation of soils, dewatering of sludges, and hazardous waste remediation among others. Estimating the feasibility of employing electro-osmosis in a particular operation depends on the ability to predict the outcome under a variety of conditions. Predictions of this type are frequently facilitated by the use of a mathematical model designed to represent the physical system under consideration in a rigorous fashion. First, a review of fundamental aspects of electro-chemico-osmotic flow in soils is presented. Following a brief outline of previous studies, identification and quantification of the significant processes, and the construction of mathematical representations are given. This is achieved using an approach based on the macroscopic conservation of mass equations and the principle of a continuum, in contrast to an approach based on the irreversible thermodynamics of coupled flows. Special emphasis is given to coupling effects on transport processes. A complete model and associated boundary conditions are then obtained for electrokinetic processes in a compressible porous medium. The proposed model takes into consideration the migration of a contaminant plume in a flow field generated by an applied electric potential.Symbols a _v soil compressibility - A an entity - C _w mass fraction of water component in the water phase - C _s mass fraction of chemical component in the water phase - C ^* capacitance of the porous medium per unit volume of porous volume - D mechanical dispersion coefficient - D _fw ^ps hydrodynamic diffusion tensor for the chemical component in the water phase - D _fw ^pw hydrodynamic dispersion coefficient for the water component in the water phase - D _f( )/Dt material derivative with respect to an observer moving at the water phase velocity V _f - D _s( )/Dt material derivative with respect to moving solids - e void ratio - f a function - F = 0 equation of a moving boundary - g gravitational acceleration - k permeability tensor of the porous medium - k _e coefficient of electro-osmotic permeability - k _ec coefficient of migration potential - k _hc chemico-osmotic coupling coefficient - m _i number of moles of the ith component - m _i0 number of moles of the ith component at a reference level - n porosity - p pore pressure - p _oi pore pressure at a reverence level - q specific discharge of water phase - q _e current density - q _fe ^p0 constant current density applied at a boundary - q ₀ constant flow rate - q _r specific discharge of the water phase relative to the moving solid matrix - R net mass transfer rate of the chemical component in the water phase - t time - u velocity of a moving surface - _i partial molar density of ith component - V _f velocity of the water phase - V _s velocity of the solid (rate of deformation) - x vertical coordinate - coefficient of matrix compressibility - _p compressibility of water phase in motion - total (overburden) stress tensor - effective stress tensor - _h streaming current conductivity - _e electrical conductivity - electrical potential - _f viscosity of the water phase - _hf density of the water phase     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

A model is presented describing the changes that occur in the diffusion boundary layer upon injection of a macromolecular solution (PEO) into a cylindrical pipe under turbulent flow conditions (Re 40,000). A shape parameter was introduced to describe the shape of the turbulent plume. The value of this parameter was found to be the same for water and various dilute PEO solutions. The proposed model gives a good approximation at low homogeneous concentrations. x downstream distance from the slot - y normal distance from the wall - R radius of the pipe - C concentration - C _w wall concentration - Q _i flow rate injection - Q _t flow rate - C _j =C _i *Q _i /Q _t equivalent homogeneous polymer concentration - L _tf characteristic length of the diffusion plume - characteristic height of the diffusion plume, i.e. the value ofy at whichC/C _w = 0.5 - thickness of the diffusion boundary layer - x ₀ characteristic distance from the slot, i.e. the value ofx at which/R = 1/2 - ⁺ shape parameter of the diffusion boundary layer - ⁺ —/R nondimensionalized variables - x ⁺ —x/L _tf nondimensionalized variables     

On slip effect in free coating of non-Newtonian fluids

The failure of the current theories to predict the coating thickness of non-Newtonian fluids in free coating operations is shown to be a result of the effective slip at the moving rigid surface being coated. This slip phenomenon is a consequence of stress induced diffusion occurring in flow of structured liquids in non-homogeneous flow fields. Literature data have been analysed to substantiate the slip hypothesis proposed in this work. The experimentally observed coating thickness is shown to lie between an upper bound, which is estimated by a no-slip condition for homogeneous solution and a lower bound, which is estimated by using solvent properties. Some design considerations have been provided, which will serve as useful guidelines for estimating coating thickness in industrial practice.fa exponent in eq. (15) - b n/(4 –n)(n + 1) - Ca Capillary number - D diffusivity - De Deborah number - g acceleration due to gravity - G Goucher number - h thickness profile - h ₀ final coating thickness - K consistency index - L length available for diffusion - L _t tube length - n power-law index - P pressure drop - Q flow rate - R cylinder radius - R _t tube radius - t time available for diffusion - T ₀ dimensionless thickness without slip - T _s dimensionless thickness with slip - U _c theoretically calculated withdrawal velocity to match the film thickness - u _s slip velocity - U withdrawal velocity - U _w theoretically calculated withdrawal velocity based on solvent properties - U ^* effective withdrawal velocity - x distance in the direction of flow - y distance transverse to the flow direction - curvature coefficient - slip coefficient - curvature coefficient - rate of deformation tensor - u _s /U - relaxation time - density - surface tension - shear stress in tube flow - _w wall shear stress in tube flow - stress tensor - _w wall shear stress - T _s /T ₀ NCL-Communication No. 2818     

Control of low-speed turbulent separated flow using jet vortex generators

A parametric study has been performed with jet vortex generators to determine their effectiveness in controlling flow separation associated with low-speed turbulent flow over a two-dimensional rearward-facing ramp. Results indicate that flow-separation control can be accomplished, with the level of control achieved being a function of jet speed, jet orientation (with respect to the free-stream direction), and jet location (distance from the separation region in the free-stream direction). Compared to slot blowing, jet vortex generators can provide an equivalent level of flow control over a larger spanwise region (for constant jet flow area and speed).Nomenclature C _p pressure coefficient, 2(P-P_)/V ² - C _Q total flow coefficient, Q/ v - D ₀ jet orifice diameter - Q total volumetric flow rate - R Reynolds number based on momentum thickness - u fluctuating velocity component in the free-stream (x) direction - V free-stream flow speed - VR ratio of jet speed to free-stream flow speed - x coordinate along the wall in the free-stream direction - jet inclination angle (angle between the jet axis and the wall) - jet azimuthal angle (angle between the jet axis and the free-stream direction in a horizontal plane) - boundary-layer thickness - momentum thickness - lateral distance between jet orifices A version of this paper was presented at the 12th Symposium on Turbulence, University of Missouri-Rolla, 24–26 Sept. 1990     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

Hydrodynamical changes due to polymer migration in very dilute solutions

The slip hypothesis, based on thermodynamical arguments, has been extended to obtain the flow characteristics of polymer solutions flowing in a nonhomogeneous flow field. An asymptotic analysis, valid for both channel and falling film flows, is presented that predicts the flow enhancement due to polymer migration. Concentration-viscosity coupling is shown to be a critical factor in the hydrodynamic analysis. The analysis, which essentially provides an upper bound on flow enhancement, explicitly accounts for the influence of wall shear stress, initial polymer concentration etc. A comparison with the pertinent experimental data shows reasonable agreement. c concentration - c ₀ concentration in shear-free region - c _i initial concentration - d rate of deformation tensor - g acceleration due to gravity - g ₁ function defined in eq. [13] or [15] - g ₂ function defined in eq. [18] or [20] - H half-channel thickness or film thickness - K gas law constant - L length of the channel or film - q flow rate per unit width - q ^* normalized flow rate - T temperature - v velocity - V mean velocity - y transverse distance - y _c location of solvent layer - _w /µ _s - _w /c ₀ KT - /t convected derivative - dimensionless cenentration,c/c ₀ - _c dimensionless interface concentration - _w dimensionless wall concentration - relaxation time - µ _eff effective viscosity - µ _s solvent viscosity - dimensionless transverse distance,y/H - _c dimensionless interface location - density - stress tensor - _w wall shear stress - c _i KT/ _w - ns no slip NCL-Communication No. 3155     

The response of the level of a liquid fluidized bed to a sudden change in the fluidizing velocity

Summary When the fluidizing velocity in a liquid fluidized bed of solid particles is suddenly changed, a discontinuity in the porosity is introduced at the bottom of the bed. This discontinuity is propagated upwards through the bed. The boundary between the old and the new porosity broadens or remains sharp depending on whether the porosity is increased or decreased. This behaviour is reflected in the way in which the bed level changes as a function of time. For a few different systems such response curves have been measured by means of a specially designed follow-up system. On the basis of the above mechanism a quantitative theory was developed for the response of the bed level to a step-wise change in the fluidizing velocity. This theory proved to give a satisfactory agreement with the observed facts.List of symbols m.k.s. units have been used for the purpose of calculation - d _p diameter of particle - h instantaneous height of the fluidized bed - n constant in eq. (2) - t time - u average velocity of the fluidized particles with respect to the wall, positive in the direction of the liquid flow - U _s settling velocity of single particle in tube; constant in eq. (2) - v average liquid velocity with respect to the wall - w() velocity of propagation of a disturbance d at a porosity - x coordinate in the direction of flow - porosity, void fraction - _p particle density - liquid velocity in region above particles, volumetric flow per unit area of empty tube - index 0 refers to the steady situation for t 0 - index 1 refers to the steady state situation state reached after t = t ₁     

Thermal radiation effects on MHD-Rayleigh flow with constant surface heat flux

Thermal radiation heat transfer effects on the Rayleigh flow of gray viscous fluids under the effect of a transverse magnetic field are investigated. The free convection heat transfer problem from constant surface heat flux moving plate is selected for study. It is found that the increasing of the magnetic field number M= H₀² / U₀²decreased velocities inside boundary layer, the increasing of the conduction–radiation parameter Rd=k__R/4aT³ decreased both temperatures and heat transfer rates. It is also found that the increasing of the dimensionless surface heat flux parameter q₀^*=q₀ /(kU₀T₎ increased the temperatures inside the boundary layer and increased the heat transfer rates. Comparison with previous works shows excellent agreement. Different transient velocity profiles, temperature profiles and local Nusselt numbers against different dimensionless groups are drawn.     

Self-similar solutions for displacement of non-Newtonian fluids through porous media

This paper presents a class of self-similar solutions describing piston-like displacement (single-phase flow is included as a special case) of one slightly compressible non-Newtonian, power-law, dilatant fluid by another through a homogeneous, isotropic porous medium. These solutions can be used to evaluate the validity and accuracy of existing approximate solutions, such as the assumption of constant flow rate at each radial distance that Ikoku and Ramey use to linearize the partial differential equation for the flow of non-Newtonian, power-law fluid through a porous medium.Nomenclature a parameter, defined by (A8) - A cross-section area of linear reservoir - B constant - c fluid compressibility - c _f formation compressibility - c _t system compressibility - c t dimensionless system compressibility, defined by (24) - C constant of integration - D _I dimensionless coefficient, directly proportional to injection rate, for linear displacement case, defined by (22). - D ₂ dimensionless coefficient, directly proportional to injection rate, for radial displacement case, defined by (55) - erf(x) error function - ercf(x) complementary error function - Ei(x) exponential integral - f dimensionless pressure, defined by (10) - h formation thickness - k permeability - l linear location of moving boundary between the displacing and displaced fluids - n flow behavior parameter - p pressure - p _i injection pressure - p ₀ initial pressure; reference pressure - p ₀ dimensionless initial pressure, defined by (19) - q injection rate - r radial distance - R radial location of moving boundary between the displacing and displaced fluids - t time - u superficial velocity - U substitution of variable - x linear distance - _e effective viscosity - _e dimensionless effective viscosity, defined by (24) - dimensionless variable, defined by (9) or (45) - _i0 value of corresponding to the location of the moving boundary between the displacing and displaced fluids - density - ₀ value of density at reference pressure - porosity - ₀ value of porosity at reference pressure - 1 displacing fluid - 2 displaced fluid     

Berechnung der Strömung des Tandemgitters mit bewegter zweiter Schaufelreihe

Ohne ZusammenfassungBezeichnungen. Es sollen bedeuten l die Profiltiefe - t die Teilung - s die Spaltweite - =+i komplexe Koordinaten der physikalischen Strömungsebene - z=x+iy komplexe Koordinaten der doppelperiodischen Strömungsebene - ¯w=u–iv die komplexe Geschwindigkeit - U die Geschwindigkeit, mit der sich das zweite Gitter bewegt - w _n die Normalkomponente der Geschwindigkeit - w _t die Tangentialkomponente der Geschwindigkeit (w _n und w _t sind reelle Größen) - Q die Quellstärke - die Wirbelstärke - q die Quellverteilung     

Rheological behaviour of two-phase flow of solid particles in a gas

The present investigation was concerned with the rheological behaviour of dilute suspensions of solid particles in a gas in a vertical cocurrent flow moving upwards. Starting from the experimentally determined dependence of the pressure drop on the concentration of solid particles and the Reynolds number of the carrier medium in the steady flow region, the rheological parameters were estimated using pseudo-shear diagrams. Air was the carrier medium and the dispersed phase was one of six fractions of polypropylene powder and five fractions of glass ballotini. The results show that the investigated two-phase systems have pseudoplastic character which becomes more pronounced with increases in concentration, equivalent diameter and density of solid particles in the flowing suspension. C _d coefficient of particle resistance - d _e equivalent diameter of particles - D column diameter - Fr Froude number - g gravitational acceleration - K rheological parameter - L length - n rheological parameter - p _t pressure drop due to friction - p _m total pressure drop - p _ag pressure drop due to acceleration of the gas phase - p _as pressure drop due to acceleration of the solid phase - p _g hydrostatic pressure of the gas phase - p _s specific effective weight of the dispersed phase - r radius - Re Reynolds number - Re _p Reynolds number of a particle - Re _G generalized Reynolds number - Re _G1 generalized Reynolds number relating to the end of the laminar flow region - Re _G2 generalized Reynolds number relating to the beginning of the turbulent flow region - w _z axial component of velocity - u _t steady free-fall velocity of a single particle - w average velocity - w _g average velocity of the gas phase - w _s average velocity of the dispersed phase of solid particles - relative mass fraction of solid particles - x _s volume fraction of solid particles - _g coefficient of pressure drop due to friction - µ dynamic viscosity - _g density of the gas phase - _m density of the suspension - _s density of solid particles - _ds density of the dispersed phase - _w shear stress at the wall     

Flow of rarefied gas past a sweptback cylinder

A numerical investigation has been made of the hypersonic flow of a rarefied monatomic gas past the windward part of the side surface of an infinite circular cylinder. The calculation was made by direct statistical Monte Carlo modeling for freestream Mach number M_t8=20, ratio of the surface temperature of the body to the stagnation temperature equal to t_tw =T _tw/T _t0 = 0.03, sweep angle 75°, and Reynolds number Re_t0 30.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1992.     

Stoffübergang mit chemischer Oberflächenreaktion an umströmten Platten

Zusammenfassung Die Strömung und der Stofftransport in der Umgebung von Platten mit chemischer Oberflächenreaktion lassen sich durch Differentialgleichungen zuverlässig beschreiben. Deren vollständige Lösung konnte ohne vereinfachende Annahmen mit Hilfe theoretisch-numerischer Methoden erzielt werden. Dadurch erhält man Einblick in die tatsächlichen Transportvorgänge. Einige wichtige Ergebnisse werden erörtert. Insbesondere wird ein umfassendes Gesetz für den Stoffübergang mitgeteilt, das theoretisch und experimentell einwandfrei gesichert ist. Die Wiedergabe der bekannten sowie der neuen Daten ist gut. Sein Gültigkeitsbereich ist angegeben. Das neue Gesetz enthält neben anderen Grenzgesetzen auch das auf der Grundlage der GrenzschichtHypothese aufgestellte Gesetz.

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Mass transfer with chemical surface reaction on flat plates in flow

The flow field and mass transfer from flat plates with chemical surface reaction can be described by means of differential equations. Their solutions have been obtained numerically without any simplifications. This report presents some of the more important results obtained, which give insight into the true transport phenomena.A comprehensive mass transfer law has been developed, that has a wide range of validity. It is in good agreement with all available experimental and theoretical data. The new mass transfer equation includes the special case of boundary layer law besides other special laws that describe mass transfer in limited regions of relevant parameters.

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Formelzeichen c_A örtliche Moldichte der reagierenden Komponente A - c_Aw Wert von c_A an der Plattenoberfläche - c Funktion nach Gl. (28) - D Diffusionskoeffizient - f_p Funktion nach Gl.(2) - k Funktion nach Gl.(27) - k_w Reaktionsgeschwindigkeitskonstante - L Länge der Platte - n Reaktionsordnung - n_A Molstromdichte der diffundierenden Komponente A - p Funktion nach Gl.(29) - r_A Reaktionsstromdichte der reagierenden Komponente A - Sh_x,Sh örtliche und mittlere Sherwood-Zahl - w Anströmgeschwindigkeit des Fluidgemisches - w_x, w _x ^* absolute und bezogene örtliche Längsgeschwindigkeit - w_y, w _y ^* absolute und bezogene örtliche Quergeschwindigkeit - x, x^* absolute und bezogene Längskoordinate - y, y^* absolute und bezogene Querkoordinate - _x, örtlicher und mittlerer Stoffübergangskoeffizien - dynamische Viskosität des Fluidgemisches - Massendichte des Fluidgemisches - Da k_wLc ^n–1 /2D Damköhler-Zahl - Re wL//gr Reynolds-Zahl - Re_kr=5 · 10⁵ kritischer Wert der Reynolds-Re_kr=5 · 10⁵ Zahl - Sc //D Schmidt-Zahl - c_A/c_A bezogene örtliche Konzentration - _w Wert von an der Plattenoberfläche Indizes A diffundierende und reagierende Komponente - w an der Plattenoberfläche - x in Längsrichtung - y in Querrichtung - in sehr großer Entfernung von der Platte     

An experimental study on vortex breakdown in a differentially-rotating cylindrical container

The vortex breakdown phenomenon in a closed cylindrical container with a rotating endwall disk was reproduced. Visualizations were performed to capture the prominent flow characteristics. The locations of the stagnation points of breakdown bubbles and the attendant global flow features were in excellent agreement with the preceding observations. Experiments were also carried out in a differentially-rotating cylindrical container in which the top endwall rotates at a relatively high angular velocity _t, and the bottom endwall and the sidewall rotate at a low angular velocity _sb. For a fixed cylinder aspect ratio, and for a given relative rotational Reynolds number based on the angular velocity difference _t–_sb, the flow behavior is examined as |_sb/_t| increases. For a co-rotation (_sb/_t>0), the breakdown bubble is located closer to the bottom endwall disk. However, for a counter-rotation (_sb/_t<0), the bubble is seen closer to the top endwall disk. For sufficiently large values of _sb, the bubble ceases to exist for both cases.     

Saint-Venant's principle in linear two-dimensional elasticity for non-striplike domains

Let be a simply connected domain in the x ₁-x ₂ plane which lies within the strip 0<x ₂, is a simple closed piecewise smooth curve. Let l= [(x ₁, x ₂): (x ₁, x ₂) and x ₁>0], _l = [(x ₁ x ₂): (x ₁ ,x ₂) and x ₁>1>0].Suppose that a two-dimensional homogeneous isotropic elastic body occupies , that a self-equilibrated stress loading is applied to - l, and that l is stress-free. Knowles [2] and Flavin [6] showed that the elastic energy in _l decays exponentially with respect to l with an exponential decay constant of the form k/b, where k is a universal constant. It is shown here that a decay constant of the form c/ may be obtained where c is a universal constant and is a characteristic dimension of , which is more appropriate than b for general non-striplike domains. In addition, an appropriate decay theorem is obtained for coil-like domains.     

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     

The mathematical modelling of thermal radiation in high temperature fluidized bed

The aim of this study is composed of two parts. One of them is to calculate the radiation heat flux and the other is to determine the overall heat transfer coefficient for the gas-fluidized bed. The radiative heat transfer model is developed for predicting the total heat transfer coefficients between submerged surfaces and fluidized beds for several working temperatures. The role of radiation heat transfer in the overall heat transfer process at an immersed surface in a gas-fluidized bed at high temperatures is investigated. Analytical results are compared with the previously done experiments and a good agreement between the two, is obtained.

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Bestimmung der Wärmeübertragungs-Koeffizienten in Gas-Wirbelschichten

Zusammenfassung Diese Untersuchung besteht aus folgenden zwei Teilen: 1. Kalkulation des Radiationswärmeübergangs in Gas-Wirbelschichten. 2. Bestimmung des Wärmeübergangs-Koeffizienten in Gas-Wirbelschichten. Dieses Radiationswärmeübergangsmodell wurde entwickelt, um die Wärmeübertragungs-Koeffizienten zwischen der eingetauchten Oberfläche und der Wirbelschicht bei verschiedener Wärme schätzungsweise zu bestimmen. Es wurde das Verhältnis der Radiationswärmeübertragung in Gas-Wirbelschichten zum totalen Wärmeübergang untersucht. Die Meßwerte wurden mit theoretischen Resultaten verglichen.

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Nomenclature c (x) specific heat capacity of packet [J/kg K] - c _p specific heat capacity of particle [J/kg K] - c _pg specific heat capacity of gas [J/kg K] - d _p average diameter of the bed particles [m] - f ₀ the fraction of time that a unit surface exposed to the bubble phase - 1–f ₀ the fraction of time that a unit surface exposed to the packet phase - g acceleration due to gravity [m/s²] - h _b heat transfer coefficient for the surface in contact with bubble [W/m² K] - h _bc conduction heat transfer coefficient for the surface/bubble [W/m²K] - h _br radiation heat transfer coefficient for the surface/bubble [W/m²K] - h _p heat transfer coefficient for the surface in contact with packet [W/m²K] - h _pc conduction heat transfer coefficient for the surface/packet [W/m² K] - h _pr radiation heat transfer coefficient for the surface/packet [W/m² K] - h _T total heat transfer coefficient between bed and surface [W/m² K] - k ⁰ thermal conductivity of the emulsion phase for fixed bed [W/m K] - k(x) thermal conductivity of packet [W/m K] - k _e the logarithmic mean of conductivity for first layer in packet [W/m K] - k _g the logarithmic mean of conductivity for the first layer in packet [W/m K] - K extinction coefficient [1/m] - m mass [kg] - n number of layers - p air pressure [pa] - q _pc mean local conduction heat transfer for packet [kW/m²] - q _pr mean local radiation heat transfer for packet [kW/m²] - Q _p average heat flux during packet contact with surface [kW/m²] - Q _b average heat flux during bubble contact with surface [kW/m²] - R gas constant [287.04 J/kg K] - t time [s] - t _g residence time for gas bubble [s] - t _k residence time for packet [s] - T temperature [K] - T _b bed temperature [K] - T _W surface temperature [K] - V _mf minimum fluidization velocity [m/s] - v _t terminal velocity [m/s] - x distance [m] Greek symbols t time increment - x thickness of the layer - emissivity - thermal diffusivity [m²/s] - (x) voidage of fluidized bed - _mf void ratio of the bed at minimum fluidization - ₀ voidage of fixed bed - _g dynamic viscosity of gas [kg/m s] - _g kinematic viscosity of gas [m²/s] - (x) density of packet [kg/m³] - _p density of particles [kg/m³] - _g density of gas [kg/m³] - Stefan-Boltzmann constant [5.66·10^–8 W/m²K⁴] - geometric shape factor for particles Dimensionless numbers Ar Archimedes numberAr=g d _p ³ ( _p – _g ) _g / _g ² - Nu Nusselt numberNu=h·d/k - Re Reynolds numberRe=d _p ·V _mf / _g - Pr Prandtl numberPr=C _{pg g} /k _g      

Interactions in general continua with microstructure

This paper studies the behavior of the one dimensional Broadwell model of a discrete three velocity gas on a bounded domain 0 x 1 with specularly reflective boundary condition at x = 0, 1. For smooth initial data we show that the initial boundary value problem possesses a unique smooth solution which tends as t to a free state consisting of traveling waves f _1e (x – ct), f _2e (x + ct), f _3e (x) where each f _ie is 2-periodic. The convergence is in the weak* topology of an appropriate Orlicz-Banach state space. No smallness assumptions are made on the data.In memory of Ronald J. DiPerna     

Heat transfer between gas-liquid spray stream flowing perpendicularly to the row of the cylinders

Experimental investigation and analysis of heat transfer process between a gas-liquid spray flow and the row of smooth cylinders placed in the surface perpendicular to the flow has been performed. Among others, there was taken into account in the analysis the phenomenon of droplets bouncing and omitting the cylinder as well as the phenomenon of the evaporation process from the liquid film surface.In the experiments test cylinders were used, which were placed between two other cylinders standing in the row.From the experiments and the analysis the conclusion can be drawn that the heat transfer coefficients values for a row of the cylinders are higher than for a single cylinder placed in the gasliquid spray flow.

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Wärmeübergang an eine senkrecht anf eine Zylinderreihe auftreffende Gas-Flüssigkeits-Sprüh-Strömung

Zusammenfassung Es wurden Messungen und theoretische Analysen des Wärmeübergangs zwischen einer Gas-FlüssigkeitsSprüh-Strömung und den glatten Oberflächen einer Zylinderreihe durchgeführt, die senkrecht zum Sprühstrahl angeordnet waren. Dabei wurde in der Analyse unter anderem das Phänomen betrachtet, daß die Tropfen die Zylinderwand treffen und verfehlen können und daß sich ein Verdampfungsprozeß aus dem flüssigen Film an der Zylinderoberfläche einstellt.Gemessen wurde an einem zwischen zwei Randzylindern befindlichen Zylinder.Die Experimente und die Analyse gestatten die Schlußfolgerung, daß der Wärmeübergangskoeffizient für eine Zylinderreihe höher ist als für einen einzelnen Zylinder in der Sprühströmung.

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Nomenclature a distance between axes of cylinders, m - c _l specific heat capacity of liquid, J/kg K - c _g specific heat capacity of gas, J/kg K - D cylinder diameter, m - g _l mass velocity of liquid, kg/m²s - ¯k average volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - k() local volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - L specific latent heat of vaporisation, J/kg - m mass fraction of water in gas-liquid spray flow, dimensionless - M constant in Eq. (9) - p pressure, Pa - p _g statical pressure of gas, Pa - p _w pressure of gas on the cylinder surface, Pa - p external pressure on the liquid film surface, Pa - r cylindrical coordinate, m - R radius of cylinder, m - T temperature, K, °C - T _l, t_l liquid temperature in the gas-liquid spray, K, °C - T _w,t_w temperature of cylinder surface, K, °C - T temperature of gas-liquid film interface, K - U liquid film velocity, m/s - w gas velocity on cylinder surface, m/s - w _g gas velocity in free stream, m/s - W _l liquid vapour mass ratio in free stream, dimensionless - W liquid vapour mass ratio at the edge of a liquid film, dimensionless - x coordinate, m - y coordinate, m - z complex variable, dimensionless - average heat transfer coefficient, W/m²K - local heat transfer coefficient, W/m² K - average heat transfer coefficient between cylinder surface and gas, W/m² K - _g, local heat transfer coefficient between cylinder surface and gas, W/m² K - mass transfer coefficient, kg/m²s - liquid film thickness, m - _lg dynamic diffusion coefficient of liquid vapour in gas, kg/m s - pressure distribution function on a cylinder surface - function defined by Eq. (3) - _l liquid dynamic viscosity, kg/m s - _g gas dynamic viscosity, kg/m s - cylindrical coordinate, rad, deg - _l thermal conductivity of liquid, W/m K - _g thermal conductivity of gas, W/m K - mass transfer driving force, dimensionless - _l density of liquid, kg/m³ - _g density of gas, kg/m³ - _w shear stress on the cylinder surface, N/m² - _w shear stress exerted by gas at the liquid film surface, N/m² - air relative humidity, dimensionless - T -T _w - _w =T _wT_l Dimensionless parameters I= enhancement factor of heat transfer - m ^*=M _l/M_g molar mass of liquid to the molar mass of gas ratio - Nu _g= D/ _g gas Nusselt number - Pr _g=c _{g g}/_g gas Prandtl number - Pr _l=c_lll liquid Prandtl number - ¯r=(r–R)/ dimensionless coordinate - Re _g=w_gD _g/_g gas Reynolds number - Re _g,max=w_g,max D _g/_g gas Reynolds number calculated for the maximal gas velocity between the cylinders - Sc=m ^* _g/_l–g Schmidt number =/R dimensionless film thickness     

Certain yawed magnetogasdynamic flows

Certain steady yawed magnetogasdynamic flows, in which the magnetic field is everywhere parallel to the velocity field, are related to certain reduced three-dimensional compressible gas flows having zero magnetic field. Under a restriction, the reduced flows are linked, by certain reciprocal relations, to a four parameter class of plane gas flows. In the instance of constant entropy an approximation method is suggested for obtaining magnetogasdynamic flows from the corresponding plane, irrotational gasdynamic flows and examples are given.

Nomenclature

magnetogasdynamic flow variables H magnetic intensity - q fluid velocity - fluid density - p pressure - s entropy - Q _t, H _t component of q, H in the x–y plane - w , h component of q, H perpendicular to the x–y plane reduced gasdynamic flow factor of proportionality - q* fluid velocity - * fluid density - p* pressure - Q _t ^* =u*î+v*, w* components of q* - l arbitrary constant - A _v Alfvén speed - Q _t, , p fluid velocity, density, pressure of the reciprocal gas dynamic flow - L, n, k, arbitrary constants - , velocity potential, stream function - angle made by Q _t, Q _t ^* , and V with the x-axis - adiabatic gas constant - a ²=(–1)/2 constant - M Mach number - W constant value of w* - E approximate constant value of g(p) - * modified potential function - modified velocity coordinate - +i - complex potential of the irrotational flow - B arbitrary constant - V incompressible flow velocity - V modified fluid velocity - X _p, Y _p points on the profile