Percolation approach to the problem of hydraulic conductivity in porous media

While percolation theory has been studied extensively in the field of physics, and the literature devoted to the subject is vast, little use of its results has been made to date in the field of hydrology. In the present study, we carry out Monte Carlo computer simulations on a percolating model representative of a porous medium. The model considers intersecting conducting permeable spheres (or circles, in two dimensions), which are randomly distributed in space. Three cases are considered: (1) All intersections have the same hydraulic conductivity, (2) The individual hydraulic conductivities are drawn from a lognormal distribution, and (3) The hydraulic conductivities are determined by the degree of overlap of the intersecting spheres. It is found that the critical behaviour of the hydraulic conductivity of the system,K, follows a power-law dependence defined byK (N/N _c–1)^x, whereN is the total number of spheres in the domain,N _c is the critical number of spheres for the onset of percolation, andx is an exponent which depends on the dimensionality and the case. All three cases yield a value ofx1.2±0.1 in the two-dimensional system, whilex1.9±0.1 is found in the three-dimensional system for only the first two cases. In the third case,x2.3±0.1. These results are in agreement with the most recent predictions of the theory of percolation in the continuum. We can thus see, that percolation theory provides useful predictions as to the structural parameters which determine hydrological transport processes.     

Stochastic analysis of two-phase flow in porous media: I. Spectral/perturbation approach

Stochastic analysis of steady-state two-phase (water and oil) flow in heterogeneous porous media is performed using the perturbation theory and spectral representation techniques. The governing equations describing the flow are coupled and nonlinear. The key stochastic input variables are intrinsic permeability,k, and the soil and fluid dependent retention parameter, . Three different stochastic combinations of these two imput parameters were considered. The perturbation/spectral analysis was used to develop closed-form expressions that describe stochastic variability of key output processes, such as capillary and individual phase pressures and specific discharges. The analysis also included the estimation of the effective flow properties. The impact of the spatial variability ofk and on the variances of pressures, effective conductivities, and specific discharges was examined.     

Constraints on parameters in three soil-water capillary retention equations

The constraints on the different fitting parameters used in the water-retention equations of Brooks and Corey, Brutsaert, and Van Genuchten are analyzed. It is shown that in the dry part of the h() curve all three equations might break down for clay soils when the fitting parameters are calculated by a best fit over observed field data. In that case, application of the h() equations for water transfer dealing with the very dry parts of the water retention curve can lead to erroneous results and must be limited to the range of water contents of the original data set.     

Discretized ordinary differential equations and the Conley index

LetN be a compact isolating neighborhood of an isolated invariant setK with respect to an ODEx=f(x) (C) and(h) x=x + h(x, h) be a consistent one-step-discretization of (C). It is proved in this paper that for someh ₀ > 0 and allh ]0, h₀[, the setN isolates an invariant setK(h) of(h) and the discrete Conley index ofK(h) coincides with the continuous Conley index ofK.     

Experimental and numerical results on transient heat transfer to supercritical helium at low temperatures

Transient heat transfer coefficients to a forced flow supercritical helium at low temperatures have been measured and compared with data of a numerical computer simulation. The helium flow through the cooling tubes was described in the simulation by a two dimensional model. The helium properties were stored as a function of enthalpy and pressure in look up tables.The experimental and numerical results agree well. At this moment the numerical code is a good instrument for computing the thermal hydraulic behaviour of hollow superconductors, cooled by a flow of supercritical helium, to get an impression on stability and cooling performance.

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Experimentelle und numerische Lösungen für transienten Wärmetransport von überkritischem Helium bei niedrigen Temperaturen

Zusammenfassung Es wurden transiente Wärmeübertragungskoeffizienten einer erzwungenen Strömung von überkritischem Helium bei niedrigen Temperaturen gemessen und verglichen mit Daten einer numerischen Computersimulation. Der Heliumstrom durch die Kühlrohre wurde in der Simulation von einem zweidimensionalen Modell beschrieben. Die Eigenschaften des Heliums wurden als eine Funktion von Enthalpie und Druck gespeichert. Die experimentellen und numerischen Ergebnisse stimmen gut überein. Folglich ist das numerische Verfahren ein gutes Instrument das thermisch-hydraulische Verhalten von hohlen Supra-Leitern, gekühlt von einem Strom überkritischen Heliums, zu berechnen, um einen Eindruck von Stabilitäts- und Kühlleistungen zu bekommen.

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Nomenclature A m² surface - a m²/s thermal diffusivity - c _p J/kg K specific heat - D m (hydraulic) diameter of the test tube - H J/kg enthalpy — in flow - J/m³ enthalpy — in tube - h W/m² K heat transfer coefficient - L _m m mixing length - m kg mass of the test tube - P N/m² pressure - R m radius of the tube - r m radial coordinate in flow - RRR residual resistance ratio(_e, 300 K/_e, 4,2 K) - S W/m³ source term of heat - T K temperature - t s time - U m/s axial velocity - V m/s radial velocity - x m axial coordinate in tube - y m R–r, the distance from the wall - y⁺ - Z m axial coordinate in flow - N s/m² viscosity - _T N s/m₂ turbulent viscosity - J/m K thermal conductivity - kg/m³ density - _e m specific electrical resistivity - _w N/m² wall shear stress - W heat flow     

Flow in Random Porous Media

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

Bail-Down Test Simulation at Laboratory Scale

This paper presents a comparison of hydraulic oil conductivity obtained from interpreting bail-down test data to values calculated from theory. The bail-down tests were performed at laboratory scale, on a radial portion of a circular domain filled with calibrated sand allowing hydraulic oil conductivity to be calculated using Parker’s theoretical model (Parker et al. in Water Resour Res 23(4):618–624, 1987). The bail-down tests were interpreted using the modified Bouwer and Rice (Huntley in Ground Water 38(1):46–52, 2000) and the modified Cooper methods (Beckett and Lyverse in API Interact LNAPL Guide 2:1–27, 2002). The results show that (1) both interpretation methods from bail-down test data give similar hydraulic oil conductivities, and (2) the hydraulic oil conductivities estimated from bail-down test data agree well with the hydraulic oil conductivity predicted when using the Parker theoretical model. Overall, this paper confirms that the modified Bouwer and Rice (Huntley 2000) and the modified Cooper methods (Beckett and Lyverse 2002) are valid to estimate hydraulic oil conductivity, giving realistic values despite test conditions not meeting all the assumptions and boundary conditions of each analytical solution.     

Visco-elastic properties of some homopolymers and copolymers in the glass-rubber transition region

Summary In experiments on polyvinyl acetate and a number of copolymers of ethylene and vinyl acetate with different ethylene contents it was found that the frequency at which the dielectric loss factor is maximum on the-frequency curve decreases very strongly at first to about 10^–4 c/s and then less strongly with decreasing temperature. In addition, the area of this dielectric dispersion peak is strongly dependent on temperature in a temperature region near the glass temperatureT _g determined dilatometrically. In this temperature region the width of the peak is virtually constant. The dielectric transition temperatureT _tr is defined as the temperature at which the area of the dispersion peak is equal to half the sum of the (small) area at low temperature and the maximum area above the transition temperature. This definition of a transition temperature is consequently not based on a given measuring frequency. For polyvinyl acetateT _tr T _g , as obtained from dilatometry and refractometry. Interference with the measuring results of a strong dependence of the time scale of temperature variations on the transition was not observed.From dynamic-mechanical measuring results obtained at 0.2 c/s the transition temperature can be defined as the temperature at which a break occurs in the storage modulus vs. temperature curve. This mechanicalT _tr is usually 5 to 10 degrees higher than the dielectricT _tr . The dependence of the transition temperature on the copolymer composition is analogous in both cases and some phenomenological relations can be used to describe this dependence. To this end the crystalline phase and the amorphous phase in copolymers with a high ethylene content are considered to be separate and the transition temperature is plotted against the weight fraction of ethylene in the amorphous phase. Extrapolation of the curve thus obtained gives (–42 ±5) °C as the transition temperature for a branched amorphous polyethylene. This value corresponds to theT _tr measured on low density polyethylene with a degree of branching of 20 CH₃ groups/1000 C-atoms.     

Yield stress: A time-dependent property and how to measure it

This paper reviews the different aspects of the yield stress phenomenon and attempts a synthesis of knowledge. Yield stress can be probed using constant shear stress or shear rate. The magnitude of the result depends on the time allowed to determine whether the sample has developed continuous flow or has ceased flowing. It is closely associated with creep, stress growth and thixotropic breakdown and recovery, and the characteristic times of these transient responses play a part in yield stress measurement. In thixotropic fluids, yield stress is a function of structure and hence of time. In simple thixotropy, the yield stress derived from the equilibrium flow curve is the same as that for the fully built-up structure. But in many materials, the static yield stress obtained after prolonged rest is much higher than the dynamic yield stress from the equilibrium flow curve. This is associated with the phenomenon in which the equilibrium flow curve bends upwards as the shear rate is reduced to very low values. The paper also reviews the many methods that can be used to measure yield stress. It is pointed out that the choice of observation time or shear rate to use should be related to the characteristic time of the flow process to which the result is to be applied. Examples discussed are solids suspension capability of fluids, levelling and sagging, pipeline flow and start-up power requirement of mixers. CS constant structure - D diameter of Gun Rheometer tube - EFC equilibrium flow curve - G measured torque - L length of Gun Rheometer tube - P _min minimum pressure to cause flow - t time - form factor for shear stress - - _y - shear rate - a particular value of shear rate - reference shear rate - test shear rate - shear stress - _y yield stress - _yd dynamic yield stress - _ys static yield stress - 0 initial value after speed change - e equilibrium Paper presented at the British Society of Rheology Conference on New Techniques in Experimental Rheology, University of Reading, 9–11 September 1985.     

Steady-State Source Flow in Heterogeneous Porous Media

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G _d (x). The mean head G _d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function (x) and for any dimensionality d of the flow. It is obtained as a product of the solution G _d ⁽⁰⁾(x) for source flow in unbounded domain of the mean conductivity K _A and the correction _d (x) which depends on the medium heterogeneous structure. The correction _d is evaluated for a few cases of interest.Simple one-quadrature expressions of _d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving ₃ as function of the distance from the source x and of the azimuthal angle . Its dependence on x, on the particular (x) and on the anisotropy ratio is illustrated in the plane of isotropy (=0) and along the anisotropy axis ( = /2).The head factor k ^* is defined as a ratio of the head in the homogeneous medium to the mean head, k ^*=G _d ⁽⁰⁾/G _d= _d ^–1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim _x0 k ^*(x) = K _H/K _A and lim _x k ^*(x) = K ^efu/K _A, where K _A and K _H are the arithmetic and harmonic conductivity means and K ^efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of (x) principal directions the limit values of k ^* are obtained as . These values differ from the corresponding components of the effective conductivities tensor for uniform flow for = 0 and /2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.     

Formulation of the inverse axisymmetric problem of steady rotational flow of an ideal incompressible fluid in turbomachinery

The problem of the design of rotor blades within the framework of the hypothesis of an infinitely large number of blades reduces to the solution of an inverse axisymmetric problem. In the Bauersfeld-Voznesenskii formulation [1–3] this problem may be stated as follows: for given meridional flow and angular rotor velocity, construct the blade surface S ₂ (Fig. 1) passing through a given inlet ab (or exit cd) edge and a given line of intersection ad of the blade with one axisymmetric stream surface of the given meridional flow. Henceforth, S ₂ will be used to designate the median (or concave) surface of the blade, which, under certain conditions, coincides with the median interblade stream surface abcd. In [1–4], in solving this problem, use is made of the condition of coplanarity of the streamline elementd r=dr, rd,dz located on the blade surface S ₂ the relative velocity vector w and the absolute vorticity vector ×c In [5, 6] it is shown that this condition is valid only for irrotational flow incident on the rotor; consequently, its use in [1, 2] is completely legitimate, while in [3, 4] its use is inadmissible in principle, since the the equi-velocity meridional flow (i. e., that stream in which along each normal n to the meridional streamlines s the meridional component w_s of the velocity is constant (w _s/ _n=0)) assumed therein is essentially rotational [=(×c) _u 0] in the curved channel leading to the rotor.In [7] Gravalos presents the formulation and method of solution of the inverse axisymmetric problem for any arbitrary rotational meridional flow (and not just an equi-velocity flow), but does not take into account the constriction of the flow by the rotor blades, or take note of special cases of degeneration of the order and type of the equations at the boundaries and within the region; moreover, the method of solution employed assumes reduction of the quasilinear hyperbolic equation to the normal form to permit its solution by the Picard method of successive approximations.Below, we present the mathematical formulation of the inverse axisymmetric problem for any arbitrary rotational meridional flow in which account is taken of flow constriction. Cases of degeneration of the order and type of the equations are considered, the case with formation of a line of parabolic degeneration is examined, and important practical cases of the formulation of the boundary and initial conditions (Goursat problem and mixed problems), which determine the possible forms of the inlet and exit edges are studied. The problems formulated for the quasilinear hyperbolic equation can be solved with the aid of the method of characteristics, the method of finite differences, the method of straight lines, and other numerical methods.The discussion is directly applicable to radial and axial hydraulic turbines; however, it can be applied in essentially the same form to pump impellers, hydraulic converters, and also to stationary guide vanes (=0).In conclusion, the author wishes to thank G. Yu. Stepanov for discussing this work.     

Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media

This paper presents a numerical method for simulating flow fields in a stochastic porous medium that satisfies locally the Darcy equation, and has each of its hydraulic parameters represented as one realization of a three-dimensional random field. These are generated by using the Turning Bands method. Our ultimate objective is to obtain statistically meaningful solutions in order to check and extend a series of approximate analytical results previously obtained by a spectral perturbation method (L. W. Gelhar and co-workers). We investigate the computational aspects of the problem in relation with stochastic concepts. The difficulty of the numerical problem arises from the random nature of the hydraulic conductivities, which implies that a very large discretized algebraic system must be solved. Indeed, a preliminary evaluation with the aid of scale analysis suggests that, in order to solve meaningful flow problems, the total number of nodes must be of the order of 10⁶. This is due to the requirement that x _i gl _i L _i , where x _i is the mesh size, _i is a typical correlation scale of the inputs, and L _i is the size of the flow domain (i = 1, 2, 3). The optimum strategy for the solution of such a problem is discussed in relation with supercomputer capabilities. Briefly, the proposed discretization method is the seven-point finite differences scheme, and the proposed solution method is iterative, based on prior approximate factorization of the large coefficient matrix. Preliminary results obtained with grids on the order of one hundred thousand nodes are discussed for the case of steady saturated flow with highly variable, random conductivities.     

Excess thermal noise and low-frequency oscillations during capillary flow of aqueous solutions of poly(ethylene oxide)

The excess thermal noise generated in polymer solutions through narrow capillaries is studied in detail for aqueous solutions of poly(ethylene oxide), , of varying concentration. With increasing flow rate, the excess noise level increases, the noise spectrum assuming a 1/f -form with 1.5. Within a critical flow range, distinct peaks appear in the spectrum, their frequencies being multiples of a fundamental frequency. The latter frequency (f ₀) is found to increase with the flow rate; this variation, as well as that brought about by varying concentration and capillary dimensions, can be accommodated in a single curve correlatingf ₀ with the shear rate at the capillary wall. No such correlation was found for the total noise level. The value off ₀ appeared to be determined by transversal oscillations of the liquid stream entering the capillary. Addition of small amounts of silica particles (Aerosil) led to the disappearance of the peaks in the spectrum.     

Infrarot-Thermographie durch Fenster bei niedrigen Temperaturen

Zusammenfassung Für die Messung lokaler Temperaturverteilunggen an inneren Oberflächen von Kanälen und geschlossenen Hohlkörpern bietet sich die IR-Thermographie als berührungslose Meßtechnik an. Die Temperaturverteilung an einer inneren Oberfläche kann nur durch ein IR-transparentes Fenster in der gegenüberliegenden Wand gemessen werden, wenn die Kamera außerhalb angeordnet ist. Durch Reflexion und Eigenstrahlung dieses Fensters entsteht ein Fehler der von der Kamera angezeigten lokalen Temperaturen, dessen Korrektur hier untersucht wird. Hierzu wird die von der IR-Kamera empfangene Strahlung in dem von der IR-Kamera detektiertem Spektralbereich mittels einer Strahlungsbilanz ermittelt. Der Vergleich mit konventionellen Temperaturmessungen bestätigt die Gültigkeit der für die Berechnung getroffenen Annahmen und gestattet für Temperaturen 295 K T _o 365 K eine Korrektur des Fenstereinflusses mittels einer linearen Gleichung (Kalibrierfunktion). Die Einflüsse variabler Umgebungsund-und/oder Fenstertemperaturen auf die von der Kamera anzuzeigenden Temperatur und damit die Gültigkeit der Kalibrierfunktion bei von der Kalibrierung abweichenden Umgebungsbedingungen werden mit Hilfe der Strahlungsbilanz untersucht. Weiterhin wird auf den Einfluß unterschiedlicher Aufnahmewinkel auf die von der Kamera angezeigte Temperatur bzw. auf die Kalibrierfunktion eingegangen.

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Infrared-thermographie through windows at low temperatures

Infrared thermography offers itself for non-intrusive measuring of local temperature distributions at internal surfaces of channels and hollow objects. The temperature distribution at an internal surface can only be measured through an IR-transparent window, if the IR-camera is positioned outside. Thermal radiation and reflection by this window may falsify the temperature distribution indicated by the IR-camera, the required correction is analyzed in this paper. For this study the radiation reaching the IR-camera within the detected range of wavelength is considered by a balance of radiative energy fluxes. The assumptions made to evaluate this balance are verified by comparing calculated and measured temperature values and the validity is confirmed. Results show, that linear calibration curves for measurements through a window may be applied with reasonable accuracy in a range of temperature 295 K T _o 365 K of both window and environment. The influence of both temperatures on the calibration curve has been studied by considering the balance of radiative fluxes. In addition, the influences of different angles of view of the IR-camera and of different window orientations are examined.

     

On the Discretization of Distributed-Parameter Systems with Quadratic and Cubic Nonlinearities

Approximate methods for analyzing the vibrations of an Euler--Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance ( _n ) and subharmonic resonance of order one-half ( 2 _n ), where is the excitation frequency and _n is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.     

Pressure transient response of stochastically heterogeneous fractured reservoirs

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblattet al.'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parameters and, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parameters and, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoir and, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.     

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      

Contribution to the theory of the high compression ratio gas ejector with cylindrical mixing chamber

Ways of improving the operation of a gas ejector with a high compression ratio are investigated. The conditions for obtaining the maximal compression ratio at the critical operating regime of the gas ejector are studied theoretically and experimentally with account for mixing of the supersonic injecting and subsonic ejected streams ahead of the choking section. The principles for the rational utilization of the effect of stream mixing in the ejector ahead of the choking section are indicated; the use of these principles permits a several-fold increase of the compression ratio of the supersonic ejector. A theory is given for the critical regime of the gas ejector with uniformly perforated nozzle, and the hydraulic parameters of the required wall perforationss are determined. It is shown that perforation as a hydraulic factor can improve significantly the parameters of the sonic ejector in the critical regime.The foundations of modem gas ejector theory were developed by Khristianovich [1, 2]. In these studies he established the relationship between the parameters of the flow at the end of the mixing chamber (section 3, p ₀ is the total pressure, is the reduced velocity) and the parameters of the ejecting (section 1, p ₀ ,) and the ejected (p₀₁,₁) flows with account for compressibility for the ejector with a cylindrical mixing chamber (Fig. 1a). The ejector theory [1, 2] (see also [3, 4]) is given in the hydraulic approximation: the flow at the end of the mixing chamber is assumed uniform, flow friction on the mixing chamber walls is neglected. The use of the gasdynamic functions [5–9] made it possible to obtain computational equations for the ejector in a convenient form and to extend them to the case of mixing of gases with different thermophysical properties. We note that for subsonic velocities of the ejecting and ejected flows the system of ejector equations [1, 2] is supplemented by the condition of equality of the static pressures p=P₁ at the stream contact section 1.The results of extensive experimental studies of subsonic ejectors are in good agreement with the results of this theory.For sonic or supersonic velocity of the ejecting gas (=1) the condition p=p₁ is not satisfied in the general case. Fundamental for the development of ejector theory was the establishment by Millionshchikov and Ryabinkov in 1948 of the existence of a critical operating regime of the supersonic ejector [7, 10]. They showed that the limiting operating regimes of the gas ejector for high pressure differentials ==p ₀ /p₀₁ are determined by the conditions for the choking of the ejected jet by the expanding supersonic ejecting flow. With the occurrence of the critical regime the velocity of the ejected jet at the choking section (section 2, Fig. 1a) reaches the speed of sound (=1); this limits the further increase of the pressure ratio and the ejector compression ratio =p ₀ /p ₀ for a given ejection coefficient k (k is the ratio of the ejected and ejecting gas flow rates). The relationships between these flow parameters at sections 2 and 1 supplement the system of ejector equations and permit determining its critical characteristics.Millionshchikov and Ryabinkov showed that for moderate values of the pressure ratio good agreement of the theoretical and experimental ejector characteristics are given by the assumption of constant static pressure p₂=const at section 2 (Fig. 1a).The limit of the applicability of the theory based on the condition p₂= = const, was studied experimentally by Lyzhin [10].The theory of the critical regime of the gas ejector was developed in 1953 in studies of Nikol'skii, Shustov, Vasil'ev, Taganov, and Mezhirov [10, 11]. Nikol'skii showed that the condition of constant static pressure at the choking section is not in agreement with the momentum equation.For a more rigorous theoretical determination of the critical ejector regime he proposed joining between sections 1 and 2 (Fig. 1a) the calculation of the ejecting jet using the method of characteristics and the hydraulic calculation of the ejected jet; example calculations were made by Nikol'skii and Shustov. Taganov and Mezhirov suggested a method for calculating the ejector critical regime using a linear distribution of the pressure in the supersonic ejecting jet (at the choking section 2).A simple and successful method for calculating the ejector critical regime was given by Vasil'ev, who used the hydraulic representation of the ejecting and ejected flows in the choking section; both flows are assumed uniform at section 2, the static pressures in these flows in the general case are different and are determined by the momentum equation. A similar theory for the ejector critical regime was developed independently in [12, 13], and the theory with account for the supersonic ejecting flow (ahead of the choking section) was developed using the method of characteristics in [14].It should be noted that the results of the calculations of the critical characteristics of the ejectors using all three of these methods were practically indentical and in good agreement with experiment for large and moderate values of the ejection coefficients. We emphasize that in the theories of the ejector critical regime the flow mixing between sections 1 and 2 is neglected.The critical regime theory imposes significant limitations on the possible characteristics of the gas ejector, first of all, on the achievable compression ratio =p ₀ /p ₀ . Thus, from the data of [10], even for a pressure ratio =1000 the maximal theoretical value of the compression ratio for the supersonic ejector does not exceed 40 (see in Fig. 2 the limiting ejector characteristics based on the critical regime theory); for the sonic air ejector (=1) the theoretical value of 3.5 (see Fig. 9b on p. 26). Therefore it is important to analyze the methods for influencing the critical regime parameters in order to determine ways to improve the operation of the gas ejector with a high compression ratio.     

An alternative approach to the linear theory of viscoelasticity and some characteristic effects being distinctive of the type of material

In the introduction some postulates on which the linear theory of viscoelasticity is based are recalled, and the postulate of passivity is substituted by a stronger postulate called detailed passivity.Next, a symmetric formulation of this theory is presented which is founded in a well-balanced way on the limiting properties of elasticity and viscosity. This leads to the introduction of the basic functions of creep compliance J ⁺(t) and stressing viscosity ⁺(t) associated to one another, whereas the basic functions retardation fluidity ⁺(t) and relaxation modulus G ⁺(t) emerge as their time derivatives. Correspondingly, four complex basic functions are defined as their Carson transforms.In addition to the proper retardation and relaxation terms, these basic functions contain the non-disappearing constants of either instantaneous compliance J ₀ or instantaneous viscosity ₀ and also of either ultimate fluidity or ultimate modulus G . Therefrom ensues a classification of linear viscoelastic materials into four types: instantaneous elasticity or viscosity is allowed to combine with ultimate viscosity or elasticity. The latter alternative, signifying fluidlike or solidlike materials, leads, of course, to a quite different behavior in many situations; however, remarkable distinctive features are associated to the first one as well.A few respective examples are outlined: 1) propagation of shear waves in a half-space with periodic and step-shaped excitation, 2) dissipation of work in a torsional vibration damper, and 3) shear flow between two parallel porous plates with injection and suction.Finally, materials with viscous initial behavior are defended against the notion that they be of no or almost no real significance.Delivered as a Plenary Lecture at the Fourth European Rheology Conference, Seville (Spain), 4–9 September 1994. The herein only outlined topics are taken from a recently pulished monograph (Geisekus, 1994) in which complete derivations of the results and more detailed discussions are given.Dedicated to Professor K. Walters on the occasion of his 60th birthday.     

On the boundary conditions at the macroscopic level

We study the problem of the boundary conditions specified at the boundary of a porous domain in order to solve the macroscopic transfer equations obtained by means of the volume-averaging method. The analysis is limited to the case of conductive transport but the method can be extended to other cases. A numerical study enables us to illustrate the theoretical results in the case of a model porous medium. Roman Letters _sf interfacial area of the s-f interface contained within the macroscopic system m² - A _sf interfacial area of the s-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - d _s , d _f microscopic characteristic length m - g vector that maps to _s, m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l REV characteristic length, m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for the f-phase at the f-s interface - n _e outwardly directed unit normal vector at the dividing surface - T ^* macroscopic temperature field obtained by solving the macroscopic equation (3), K - V averaging volume, m³ - V _s , V _f volume of the considered phase within the averaging volume, m³ - volume of the macroscopic system, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² Greek Letters _s , _f volume fraction - ratio of thermal conductivities - _s , _f thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding to T ^* , K - spatial deviation temperature K - error on the temperature due to the macroscopic boundary conditions, K - spatial average - ^s , ^f intrinsic phase average