A 3D LIF system for turbulent buoyant jet flows

A Laser-Induced Fluorescence (LIF) system for mapping the three dimensional tracer concentration field in turbulent flows is described. The system is particularly suited to studies of single or multiple buoyant jets discharged into unstratified and stratified flowing environments for conditions typical of wastewater discharges into surface water bodies. A laser beam is scanned through the flow and LIF images are obtained in parallel planes with a high-speed synchronized CCD camera. Refractive index matching is used to minimize refractive index variations due to local density gradients. An application to vertical round buoyant jets discharging into unstratified and stratified cross flows is presented. The three-dimensional system can obtain vastly more data than is possible with probe-based techniques and can yield far more insight into the flow and mixing processes.Abbreviations a Combined attenuation coefficient, cm^-1 - a_w Attenuation coefficient for fresh water, cm^-1 - B Buoyancy flux of buoyant jet, cm⁴/s³ - c Tracer concentration, g/l - C _salt, C _eth Salt and ethanol concentrations, g/l - d Port diameter, cm - g Acceleration due to gravity, cm/s² - Modified acceleration due to gravity at source, cm/s² - Q Volume flux of buoyant jet, cm³/s - I Image gray scale level, DN - l _a,l _Q , l _M , l _s, l _t Buoyant jet length scales, cm (Eq. 8) - L _y Distance from camera to image plane, cm - M Momentum flux of buoyant jet, cm⁴/s² - M _y Image scale factor - N Buoyancy frequency, s^-1 - u _a Ambient flow velocity, cm/s - u _j Jet exit velocity, cm/s - P Laser power, W - S ₀ Lowest dilution on the vertical center-plane through the nozzle - S _m Minimum dilution: lowest dilution in a vertical plane perpendicular to the flow - x, y, z Coordinates, cm - z _m Maximum rise height of buoyant jets, cm - z _e Equilibrium rise height of buoyant jets, cm - Image calibration constant - ₀ Effluent density, g/cm³ - _a Ambient density, g/cm³     

Extensional flow of polymer solutions through porous media

The flow behaviour of various polymer solutions of non-hydrolyzed polyacrylamide, hydrolyzed polyacrylamide, polyox and Xanthan was investigated in a plexiglass column having a succession of enlargements and constrictions, and compared with the flow behaviour and mechanical degradation of a solution of non-hydrolyzed polyacrylamide in a packed column of non-consolidated sand. The flow behaviour of this solution was found to be very similar in both the sand pack and plexiglass pore.Apart from the Xanthan solution, all other polymer solutions showed a viscoelastic behaviour in the plexiglass pore. The onset of viscoelastic behaviour, which has previously been defined using the shear rate ( ), stretch rate ( _s ) and Ellis number (E ₁), could be more precisely evaluated using a modified stretch rate (S _G). The pressure losses across the plexiglass pore for different polymer solutions of the same type were found to follow a unique curve provided the suggested group (S _G) was used, a situation which was not achieved with the other rheological parameters.The multipass mechanical degradation of the non-hydrolized polyacrylamide was tested through the sand pack against the suggested group (S _G) and Maerker's group (M _a). It was found that the loss of the solution viscoelasticity due to multipass mechanical degradation was better represented usingS _G thanM _a. A cross-sectional area (cm²) - C ^* critical concentration of polymer (ppm) - d plexiglass pore enlargement diameter - D average sand grain diameter (cm) - e equivalent width for the plexiglass pore - E ₁ Ellis number (a Deborah number) - F _R resistance factor - F _Ri resistance factor at the first pass - h height of the flow path of the plexiglass pore - K power-law constant - K _h,K _w effective permeability to hydrocarbon and water, respectively (10^–8 cm²) - M _a Maerker's group for a given porosity (s^–1) - M _ai value ofM _a at the first pass - N _D Deborah number - n power-law index - Q flow rate (cm³/s) - R capillary radius (cm) - R _g radius of gyration - S _G suggested group of rheological parameters representing a modified maximum stretch rate (s^–1) - S _Gi value ofS _G at the first pass - T _R,t characteristic time for the fluid (s) - t _s residence time (s) - V ₀ superficial velocity (cm/s) - V mean velocity of flow through a porous medium (cm/s) - average axial velocity in the enlargement section of the plexiglass pore (cm/s) - V ₁,V ₂ maximum velocity at a plexiglass enlargement neck and centre - [] intrincis viscosity - viscosity (mPa s) - _r relative viscosity (ratio of the viscosity of the polymer solution to that of the solvent) - shear rate (s^–1) - _s stretch rate (s^–1) - characteristic time for the polymer solution (s)     

The effect of leaching on mixing in porous media with a single heterogeneity

This paper presents an exploratory study of the effect of leaching on mixing in a porous medium containing a single heterogeneity to investigate the effect of the heterogeneity and time-dependent pore structure on dispersion. A percolation-convection simulation (PCS) model is used along with laboratory model experiments to study the mixing. The results show that mixing changes when the pores of the models are leached and that there is a change in regime influence during leaching. The simulation represents the mixing through a first leach for homogeneous media and for heterogeneous media with significant changes in permeability. If the pore structure is changing with time, prediction of mixing must include effects of heterogeneity and regime influence. Although the experimental results are representative of idealized laboratory sized systems they provide insight into the effects of leaching in heterogeneous media. Further the simulation may be useful on a field scale.Nomenclature b molecular weight, gm/mol - C concentration, mol/cm³ - C ₀ initial concentration, mol/cm³ - d _rms root-mean-squared distance, cm - d ₅₀ 50% grain size, cm - D channel depth, cm - f _n fraction of input tracer in effluent at time t _n - K ₁ permeability of flow field outside of heterogeneity, cm² - k ₂ permeability of heterogeneity, cm² - k _S reaction rate constant, cm/min - K _L microscopic dispersion coefficient in the longitudinal direction, cm²/sec - K _O overall dispersion coefficient in the longitudinal direction, cm²/sec - K _T microscopic dispersion coefficient in the transverse direction, cm²/sec - L length of channel, cm - n exponent for velocity - P pressure, N/M² - Pe Peclet number, Lv/K _O - P _ext local pressure outside heterogeneity, N/M² - P _int local pressure inside heterogeneity, N/M² - Q volumetric flow rate, cm³/sec - R channel half width, cm - t time, sec - W _c channel width, cm - W _c0 initial channel width, cm - v interstitial fluid velocity, cm/sec - v _k macroscopic velocity in transverse direction, cm/sec - v _y macroscopic velocity in longitudinal direction, cm/sec - v fluid velocity entering the medium, cm/sec - x _i transverse location of parcel at time t _i, cm - y _i longitudinal location of parcel at time t _i, cm - x microscopic movement in transverse direction, cm - y microscopic movement in longitudinal direction, cm Greek Letters t time increment, sec - ₀ overall dispersivity, cm - ² longitudinal variance of the distribution, cm² - porosity - _B bulk density, gm/cm³ - fractional grade of leachable material Currently with Center for Naval Analysis.     

An extended theory to predict the onset of viscous instabilities for miscible displacements in porous media

Many enhanced oil recovery schemes involve the displacement of oil by a miscible fluid. Whether a displacement is stable or unstable has a profound effect on how efficiently a solvent displaces oil within a reservoir. That is, if viscous fingers are present, the displacement efficiency and, hence, the economic return of the recovery scheme is seriously impaired bacause of macroscopic bypassing of the oil. As a consequence, it is of interest to be able to predict the boundary which separates stable displacements from those which are unstable.This paper presents a dimensionless scaling group for predicting the onset of hydrodynamic instability of a miscible displacement in porous media. An existing linear perturbation analysis was extended in order to obtain the scaling group. The new scaling group differs from those obtained in previous studies because it takes into account a variable unperturbed concentration profile, both transverse dimensions of the porous medium, and both the longitudinal and the transverse dispersion coefficient.It has been shown that stability criteria derived in the literature are special cases of the general condition given here. Therefore, the stability criterion obtained in this study should be used for a displacement conducted under arbitrary conditions. The stability criterion is verified by comparing it with miscible displacement experiments carried out in a Hele-Shaw cell. Moreover, a comparison of the theory with some porous medium experiments from the literature also supports the validity of the theory.Nomenclature c solvent concentration - C _g fractional glycerine volume - D molecular diffusion coefficient, cm²/s - D _L longitudinal dispersion coefficient, cm²/s - D _T transverse dispersion coefficient, cm²/s - g gravitational acceleration, cm/s² - h distance between the plates, cm - I _sr dimensionless scaling group - k permeability, cm² - L _x width of the porous medium, cm - L _y height of the porous medium, cm - t time, s - u velocity in thex direction, cm/s - v velocity in they direction, cm/s - V displacement velocity, cm/s - w velocity in thez direction, cm/s - z length of the graded viscosity bank, cm - eigenvalue in thex direction - eigenvalue in they direction - wave number - viscosity, poise - density, g/cc - time constant, s^-1 - porosity     

Rheologische Messungen an Kunststoff-Schmelzen mit einem neuen Rotationsrheometer

Zusammenfassung In der vorliegenden Arbeit wird ein neues Rotationsrheometer vorgestellt und über Messungen an zwei Polymethylmethacrylat-Formmassen berichtet. Bei dem Rheometer handelt es sich um ein Couette-Rheometer mit feststehendem Innenzylinder als Meßkörper. Der Meßkörper ist beidseitig eingespannt. In dem geschlossenen Meßraum können die Schmelzen bis zu einem Druck von 500 bar belastet werden.Der zeitliche Verlauf der Schubspannung in den Schmelzen wird in Abhängigkeit von Temperatur und Druck aufgezeichnet.

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Summary A new type of rotational rheometer is described, and results for two samples of polymethylmethacrylate are reported. The rheometer consists of a Couette system with fixed inner cylinder, supported at both ends for torque measurements. Pressure may be varied up to 500 bar. Shear stresses have been recorded as a function of time, temperature and pressure.

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Nomenklatur C [kp cm^–2 s^–1] Steigung der Anlaufkurve im Nullpunkt - D [kp cm rad^–1] Direktionsmoment - E ₀ [kcal mol^–1] Aktivierungsenergie der Newtonschen Viskosität - G [kp cm^–2] Schubmodul - G [—] Griffith-Zahl - l [mm] Länge des Meßkörpers - p [kp cm^–2] Druck - R _i [mm] Radius des Innenzylinders - R _a [mm] Radius des Außenzylinders - t _max [s] Zeit, bei der das Maximum in der Anlaufkurve auftritt - T [°C] Temperatur - ₀ [cm² kp^–1] Druckkoeffizient der Newtonschen Viskosität - [s^–1] Schergeschwindigkeit - ₀ [kp s cm^–2] Newtonsche Viskosität - (g cm²] Trägheitsmoment des Meßkörpers - v ₀ [s^–1] Eigenfrequenz des Meßsystems - _max [kp cm^–2] maximale Schubspannung - _st [kp cm^–2] stationäre Schubspannung Mit 7 Abbildungen und 1 Tabelle     

Das Folienblasverfahren als rheologisch-thermodynamischer Prozeß

Zusammenfassung Die bisherige Forschung hat gezeigt, daß die getrennte Untersuchung der rheologischen und der thermischen Vorgänge beim Folienblasprozeß zu keiner vollständigen Analyse des Verfahrens führt. Nachdem durch intensive experimentelle Arbeiten in den letzten Jahren der Abkühlvorgang und damit die thermischen Randbedingungen des Folienblasverfahrens geklärt wurden, ist jetzt eine derartige Analyse möglich. Die auf Grund des aufgestellten Modells für verschiedene Betriebspunkte berechneten Werte von Stützluftdruck und Abzugskraft werden mit experimentellen Ergebnissen verglichen. Dabei zeigt es sich, daß man eine gute Übereinstimmung zwischen dem errechneten und dem gemessenen Stützluftdruck unter der Annahme einer mittlerennewtonschen Viskosität erhält, die mit zunehmendem Abzugsverhältnis stark abnimmt. Die gemessene Abzugskraft läßt sich jedoch nicht ohne Berücksichtigung elastischer Effekte erklären.

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Summary In blown film extrusion till now the rheological and the thermal aspect of the process have been analysed separately. After some years of intense experimental work leading to a better understanding of the cooling conditions, a combined analysis of the process seems now possible. Based on a theoretical model, the calculated inside bubble pressures and axial tensions are compared with experimental values for different blow up and draw down ratios. Agreement between calculated and measured bubble pressures can be achieved by assuming an averageNewtonian viscosity for the process, which is strongly dependent on the draw down ratio. The measured axial tension may only be explained by taking into account elastic effects.

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A — Aufblasverhältnis - B — dimensionsloser Stützluftdruck nach Gl. [16] - b °C^–1 Temperaturkoeffizient der Viskosität - C _s Stefan-Boltzmann-Konstante - c _p — relative spezifische Wärme - c _p ^* cal/g°C spezifische Wärme - c _p0 cal/g°C spezifische Wärme bei der TemperaturT ₀ - E _s °C^–3 Strahlungskonstante - E — dimensionslose Einfriergrenze nach Gl. [23] - F p Abzugskraft - F _E p Abzugskraft an der Einfriergrenze - F _E ^el p elastischer Anteil der AbzugskraftF _E - H — dimensionsloser Wärmeübergangskoeffizient nach Gl. [22] - K dimensionslose Abzugskraft nach Gl. [15] - K _E — dimensionslose Abzugskraft an der Einfriergrenze - m g/s Massedurchsatz - p p/cm² isotroper Druck - p p/cm² Stützluftdruck - q kcal/m²h Wärmestrom - r — dimensionsloser Schlauchradius - r ^* cm Schlauchradius - r ₀ cm Düsenradius - r _E cm Radius der Schlauchfolie an der Einfriergrenze - s — dimensionslose Foliendicke - s ^* cm Foliendicke - s ₀ cm Spaltweite der Düse - s _E cm Foliendicke an der Einfriergrenze - T °C Folientemperatur - T _E °C Einfriertemperatur - T ₀ °C Masseaustrittstemperatur - V — Abzugsverhältnis - v — dimensionslose Foliengeschwindigkeit - v ^* cm/s Foliengeschwindigkeit - v ₀ cm/s Masseaustrittsgeschwindigkeit - v _E cm/s Abzugsgeschwindigkeit - W — dimensionsloses spezifisches Gewicht nach Gl. [17] - x — dimensionslose Längskoordinate - x ^* cm Längskoordinate - x _E cm Einfriergrenze - kcal/m²h °C Wärmeübergangskoeffizient - — relative Emissivität - — relative Viskosität - ^* ps/cm² Viskosität - ₀ ps/cm² Viskosität bei der TemperaturT ₀ - _s ps/cm² Scherviskosität - — relative Dichte - ^* g/cm³ Dichte - ₀ g/cm³ Dichte bei der TemperaturT ₀ Vorgetragen auf der Jahrestagung der Deutschen Rheologen vom 28.–30. April 1975 in Berlin.Mit 15 Abbildungen     

Diffusion of moisture in deformable porous media. Anisotropic fibrous materials

This paper presents a study on the deformation of anisotropic fibrous porous media subjected to moistening by water in the liquid phase. The deformation of the medium is studied by applying the concept of effective stress. Given the structure of the medium, the displacement of the solid matrix is not taken into account with respect to the displacement of the liquid phase. The transport equations are derived from the model proposed by Narasimhan. The transport coefficients and the relation between the variation in apparent density and effective stress are obtained by test measurements. A numerical model has been established and applied for studying drip moistening of mineral wool samples capable or incapable of deformation.Nomenclature D mass diffusion coefficient [L²t^–1] - e void fraction - g gravity acceleration [Lt^–2] - J mass transfer density [ML^–2t^–1] - K hydraulic conductivity [Lt^–1] - K _s hydraulic conductivity of the solid phase [Lt^–1] - K ^* hydraulic conductivity of the deformable porous medium [Lt^–1] - P pressure of moistening liquid [ML^–1 t^–2] - S degree of saturation - t time [t] - V speed [Lt^–1] - X horizontal coordinate [L] - Z vertical coordinate measured from the bottom of porous medium [L] - z z-coordinate [L] Greek Letters porosity - ₁ total hydric potential [L] - _g gas density [ML^–3] - ₁ liquid density [ML^–3] - ₀ apparent density [ML^–3] - _s density of the solid phase [ML^–3] - density of the moist porous medium [ML^–3] - external load [ML^–1t^–2] - effective stress [ML^–1t^–2] - bishop's parameter - matrix potential or capillary suction [L] Indices g gas - 1 moistening liquid - p direction perpendicular to fiber planes - s solid matrix - t direction parallel to fiber planes - v pore Exponent * movement of solid particles taken into account     

Simultaneous PIV and pulsed shadow technique in slug flow: a solution for optical problems

A recent technique of simultaneous particle image velocimetry (PIV) and pulsed shadow technique (PST) measurements, using only one black and white CCD camera, is successfully applied to the study of slug flow. The experimental facility and the operating principle are described. The technique is applied to study the liquid flow pattern around individual Taylor bubbles rising in an aqueous solution of glycerol with a dynamic viscosity of 113×10^–3 Pa s. With this technique the optical perturbations found in PIV measurements at the bubble interface are completely solved in the nose and in annular liquid film regions as well as in the rear of the bubble for cases in which the bottom is flat. However, for Taylor bubbles with concave oblate bottoms, some optical distortions appear and are discussed. The measurements achieved a spatial resolution of 0.0022 tube diameters. The results reported show high precision and are in agreement with theoretical and experimental published data.Symbols D internal column diameter (m) - g acceleration due to gravity (m s^–2) - l _w wake length (m) - Q _v liquid volumetric flow rate (m³ s^–1) - r radial position (m) - r * radial position of the wake boundary (m) - R internal column radius (m) - U _s Taylor bubble velocity (m s^–1) - u _z axial component of the velocity (m s^–1) - u _r radial component of the velocity (m s^–1) - z distance from the Taylor bubble nose (m) - Z * distance from the Taylor bubble nose for which the annular liquid film stabilizes (m) Dimensionless groups Re Reynolds number ( ) - N _f inverse viscosity number ( ) Greek letters liquid film thickness (m) - liquid kinematic viscosity (m² s^–1) - liquid dynamic viscosity (Pa s) - liquid density (kg m^–3)     

New local measurements of hydrodynamics in porous media

In this paper, a measurement system is used to carry out local hydrodynamic measurements at the pore scale of a fixed-bed reactor. It consists of four microelectrodes placed on the inner wall of four spheres mounted in tetrahedral form, thus constituting a pore of the fixed bed. Three flow regimes (laminar, inertial and turbulent-like) are identified by the analysis of the signal fluctuations (velocity gradient). The flow structures are characterised by means of the correlation (auto- and cross-) function analysis, and a closure equation required in modelling and simulation is suggested.List of symbols A_e effective area of electrode, m² - C_L concentration, mol/m³ - C_xx auto-correlation function of signal x - d_ip distance between two probes, m - d_m average structure dimension, m - d_M maximum structure size, m - D diffusion coefficient, m²/s - d_e electrode diameter, m - f frequency, s^–1 - F Faraday constant, 96,500 C/equi - F_D(L–S) liquid–solid momentum exchange term, kg/m²/s² - g gravity, 9.81 m/s² - H transfer function - I limiting current, A - L liquid flow rate, kg/m²/s - P static pressure, Pa - P_xx Power spectral density of signal x - Re_p , particle Reynolds number - S velocity gradient, s^–1 - t time, s - T_c integral coherence time, s - u_L liquid phase velocity, m/s - u_L fluctuation of liquid phase velocity, m/sGreek symbols _L fluid volume fraction - porosity - liquid kinematic viscosity, Pa s - _e number of electrons involved in the electrochemical reaction - _L liquid phase specific gravity, kg/m³ - ₀ local liquid phase specific gravity, kg/m³ - time lag (in correlation functions), s - _c coherence time, s - tortuosity - dimensionless frequency     

Bestimmung der zweiten Normalspannungsfunktion von Polymerschmelzen

Zusammenfassung In der axialen Druckströmung durch einen konzentrischen Ringspalt läßt sich aus der Druckdifferenz zwischen Innen- und Außenwand die zweite Normalspannungsfunktion quantitativ bestimmen. In dieser Arbeit werden die Bestimmungsgleichungen für die Scherviskositätsfunktion bzw. die zweite Normalspannungsfunktion hergeleitet. Insbesondere wird die in diesen Ableitungen vorausgesetzte Isothermie der Strömung überprüft.Für die experimentelle Ermittlung der Scherviskositätsfunktion bzw. der zweiten Normalspannungsfunktion wurde eine Meßapparatur (Ringspaltwerkzeug) entwickelt.Die Ergebnisse zeigen, daß die hier ermittelten Werte für die Scherviskositätsfunktion gut mit denen übereinstimmen, die in anderen Geometrien gemessen wurden.Die aus der Druckdifferenz zwischen Innen- und Außenwand des Ringspaltes zu berechnende zweite Normalspannungsfunktion wird als Funktion von Schergeschwindigkeit und Temperatur dargestellt. Ähnlich wie bei den Polymerlösungen ergeben sich auch bei Polymerschmelzen negative Werte für die zweite Normalspannungsfunktion.

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Summary In annular flow, the pressure difference between inner and outer wall can be used to determine the second normal-stress coefficient. In this paper, the equations for the shear viscosity and the second normal-stress coefficient are derived. Special consideration is given to the problem of isothermal flow.On the experimental side, an annular die has been designed for the determination of the shear viscosity and the second normal-stress coefficient.The results show, that the measured values of shear viscosity coincide with those, measured by other geometries. The second normal-stress coefficient, determined by the pressure difference between inner and outer wall of the annular die, is presented as a function of shear rate and temperature. As in the case of polymer solutions, negative values for the second normal stress coefficient are obtained for polymer melts.

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a (cm²/kps) Kenngröße aus dem Ansatz nachRabinowitsch (Gl. [4]) - b (cm²/s) Temperaturleitfähigkeit - c (cm⁶/kp³s) Kenngröße aus dem Ansatz nachRabinowitsch (Gl. [4]) - E (—) dimensionslose Zahl in Gl. [14] - F ₁,F ₂ (kps²/cm²) erste bzw. zweite Normalspannungsfunktion - F _2,0 (kps²/cm²) Wert der zweiten Normalspannungsfunktion für kleine Schergeschwindigkeiten - h (cm) Spaltweite - k (kcal/m h °C) Wärmeleitfähigkeit - l (cm) Länge des Ringspaltes - n ₂ (—) Kenngröße aus dem Ansatz nachCarreau (Gl. [5]) - Na (—) Nahme-Zahl - p (kp/cm²) Druckgradient, Definition s. Gl. [7a] - Pe (—) Péclet-Zahl - r, R (cm), (—) laufender Radius - r _a (cm) Außenradius des Ringspaltes - r _i (cm) Innenradius des Ringspaltes - S _ij (kp/cm²) Komponenten des SpannungstensorsS - T (°C) Temperatur - T ₀ (°C) Ausgangstemperaturniveau - (K) mittlere Temperatur - v _z (cm/s) Strömungsgeschwindigkeit - (cm/s) mittlere Strömungsgeschwindigkeit - (cm³/s) Volumendurchsatz - ₂ (s) charakteristische Zeit aus dem Ansatz nachCarreau (Gl. [5]) - (s^–1) Schergeschwindigkeit - (kps/cm²) Scherviskosität - k (—) Radienverhältnis - (—) dimensionslose Koordinate (() = 0) - (kp/cm²) Schubspannung - (—) laufende Koordinate - (–) laufende Koordinate - 0 Bezugszustand - a an der Außenwand - i an der Innenwand - r, z, Koordinatenrichtungen Auszugsweise vorgetragen auf der Jahrestagung der Deutschen Rheologen in Berlin vom 28.–30. April 1975.Mit 8 Abbildungen     

Rheology on the drawing zone in glass spinning

Summary Among investigations concerning the rheology of spinning materials from melt, or in other terms the problem of spinnability, glasses perform an example of fibre forming without crystallization along the spinning way and with surface tension playing an important role. Furthermore glasses show aNewtonian behaviour at least in the upper part of the drawing zone.As the absence of crystallization simplifies the formulation of the governing energy equation, on the other hand, the surface tension makes the applied motion equations quite complex to solve, above all in the two-dimensional analysis.The present paper shows that only a two-dimensional approach can give reliable results on the temperature, velocity and stress distribution in the drawing zone by a comparison of the theoretical and the experimental diameter profile of the forming fibre.The temperature profile has been obtained by a numerical solution of the energy equation, only after gaining experimentally the heat transfer coefficient. The results shown in the one-dimensional analysis cannot be applied in the opper part of the drawing zone.The velocity and stress distribution can be obtained by very complex numerical solutions in the very upper part of the drawing zone where the one-dimensional approach is shown unreliable. This can be thought an asymptotic solution of two-dimensional approach, reliable only after a certain distance of the spinning way from the exit of the nozzle.Furthermore, an analysis of the dimensionless numbers involved in the spinning phenomena brings up some information concerning the instability of the glass jet in comparison with that shown by materials as molten polymers or metals.As far as the rheological behaviour of glasses in the elongational shear rate is concerned, some conclusions can be drawn. F _r Froude numberU ₀ ² /gR₀ withg acceleration gravity (cm/sec²) - N _u Nusselt number 2Rh/Ka withh heat transfer coefficient (cal/cm² sec °C) andK _a air thermal conductivity (cal/cm sec °C) around the forming fibre - Q Volume rate of flow (cm³/sec) - r Radial distance from the central axis of the fibre (cm) - R Cross section radius of the fibre (cm) - R ₀ Inside diameter of the nozzle (cm) - t Quenching time (sec) - T _aT_s Temperature of fibre at the centre (°C) - T _i Initial temperature at the distancex = 0 (°C) - T ₀ Mean value of temperature of air surrounding the forming fibre (°C) - U ₀ Mean value of velocity of glass atx = 0 (cm/sec) - V Local velocity of fibre in the axial direction (cm/sec) - x Axial distance of the fibre from the nozzle exit (cm/sec) - W Weight rate of flow (g/minute) - W _e Weber numberU ₀ ² R₀/ - Glass surface tension (dynes/cm) - Angle between the fibre axis and the tangent to the fibre surface in ther, x plane (radiant). - v Air kinematic viscosity (cm²/sec) - Glass density (g/cm³) - Glass viscosity (poises) - _i Glass viscosity atT _i. - Maxwell relaxation time/G (sec) withG (dynes/cm²) elastic shear modulus of glass With 10 figures and 2 tables     

An approximate theoretical analysis and experimental verification of turbulent entrance region flow of drag reducing fluids

Summary Entry lengths for pipe flows of moderately drag reducing fluids are determined using momentum integral technique. It is shown theoretically that the entry lengths for drag reducing fluids could be significantly larger than the Newtonian fluids flowing turbulently under otherwise identical conditions. The experimental data from the literature bear out the theoretical calculations.

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Zusammenfassung Mit Hilfe der Impuls-Methode wird die Einlauflänge in einer Rohrströmung für Flüssigkeiten mit mäßig starker Widerstandsverminderung berechnet. Es wird vorausgesagt, daß die Einlauflänge für derartige Flüssigkeiten erheblich größer sein kann als für newtonsche Flüssigkeiten unter sonst identischen Bedingungen. Aus der Literatur entnommene experimentelle Daten bestätigen diese theoretischen Berechnungen.

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Nomenclature A ₁ Coefficient in eq. [7] - A Slope of logarithmic velocity profile - a Exponent in eq. [10] - B Intercept function for logarithmic velocity profile - De Deborah number, - f Friction factor - F Function, eq. [30] - G Function given in eq. [11] - Static pressure, dynes/cm² - q Index of power law velocity profile - R Pipe radius, cm - r Radial distance, cm - R Core radius, cm - Re Reynolds number - Axial velocity, cm/s - u _c Core velocity, cm/s - u ⁺ Dimensionless velocity, eq. [5] - u ^* Friction velocity, , cm/s - Radial velocity, cm/s - V Average velocity, cm/s - x Axial distance, cm - x _e Entry length, cm - y Distance from the wall, cm - y ⁺ Dimensionless distance, eq. [5] - y _I ⁺ Dimensionless viscous sublayer thickness - coefficient in eq. [17] - exponent of Reynolds number in eq. [17] - shear rate, s^–1 - boundary layer thickness, cm - _fl fluid relaxation time, s - µ fluid viscosity, gm/cm s - v kinematic viscosity, cm²/s - _l laminar sublayer thickness, dimensionless - fluid density, gm/cm³ - shear stress, dynes/cm² - _w shear stress at the wall, dynes/cm² - ₁, ₂, ₃, ₄ functions in eq. [27] - ~ time averaged quantities - — dimensionless quantity With 3 figures and 1 table     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

Liquid sheet and film atomization: a comparative experimental study

Liquid atomization processes are too complex to allow a purely theoretical study. Therefore experiments are necessary to quantify droplets production. In our problem, the replacement of an original complicated flow by a simpler one, i.e. liquid metal and high gas velocity by water and low air velocity, has led to a relation for the droplet diameter, thanks to dynamical similarity and order of magnitude estimates. Observation of a liquid film disruption development by high speed photography gives some informations about the mechanism of break-up in action. Granulometric measurements by video image analysis have specified the previous dimensionless relation for the mass median diameter. Measurements concern both the film and the sheet atomization, it is shown that the control of the liquid layer thickness is of major importance to control the quality of sprays.List of symbols d droplet diameter (m) - d _mm mass median droplet diameter (m) - g acceleration due to the gravity (ms^–2) - H _g , H _l gas slit width, liquid film thickness (m) - dimensionless parameters - Q ₁ = H ₁ V ₁ liquid flow rate (m²s^–1) - Reynolds number - T time(s) - V _g , V _l gas and liquid velocity (m s^–1) - W _c channel width (m) - Weber number - _g , _l gas and liquid viscosity (kg m^–1 s^–1) - _g , _i gas and liquid density (kg m^–3) - surface tension (kg s^–2) An abridged version of this paper was presented at the 6th ICLASS (Int. Conf. on Liquid Atomization and Spray Systems), Rouen, France, 18–22 July 1994     

Experimentelle Untersuchung des Wärmeübergangs an Helium I bei Blasenverdampfung in einem senkrechten Rohr

Zusammenfassung In einem senkrecht angeordneten, elektrisch beheizten Kupferrohr von 5 mm Innendurchmesser und 168 mm beheizter Länge wurde der Wärmeübergang an gesättigtes Helium I für die Siedetemperaturen 2,5; 3,4; 4,2 und 4,6 K im Bereich der Blasenverdampfung gemessen. Es wurde beachtliche Hysterese festgestellt. Die Verteilung der Temperaturdifferenz über die Rohrlänge wird gezeigt. In einer anderen Versuchsreihe wurde die kritische Heizflächenbelastung zwischen-Punkt und kritischem Punkt in Abhängigkeit von der Siedetemperatur bestimmt.

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Nucleate boiling heat transfer to helium i in a vertical tube

In a vertical electrically heated copper-tube with 5 mm i. d. and a heated length of 168 mm measurements of nucleate boiling heat transfer to saturated helium I were carried out with boiling temperatures 2.5; 3.4; 4.2; and 4.6 K. A noticeable hysteresis in heat transfer was found. The variation of temperature difference along the tube is shown. In another test series the critical heat flux as a function of boiling temperature was determined between the and the critical temperatures.

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Bezeichnungen A ₀...A ₃ Polynomkoeffizienten in Gl. (7) - a Konstante in Gl. (8), cm/s - G Konstantein Gl. (9), cm/s K^1,3 - I ₁ Meßstrom der Temperaturfühler, A - I ₂ Strom der Vergleichswiderstände für die Temperaturmessung, A - K Konstante in Gl. (1), W/mK² - m Exponent in Gl. (8) - q Heizflächenbelastung, W/cm² - q _kr kritische Heizflächenbelastung, W/cm² - R _N Vergleichswiderstand für die Temperaturmessung, Ohm - R _T Widerstand der Temperaturfühler, Ohm - r Verdampfungswärme, J/g - r ₁ Abstand der Achsen der inneren Temperaturfühler von der Rohrachse, cm - r ₂ Abstand der Achsen der äußeren Temperaturfühler von der Rohrachse, cm - r _w Radius der Heizfläche, cm - T Absoluttemperatur, K - T ₁ Temperatur eines inneren Temperaturfühlers des Versuchsrohrs, K - T ₂ Temperatur eines äußeren Temperaturfühlers des Versuchsrohrs, K - T ₃ Siedetemperatur, K - T _w Heizflächentemperatur in der Höhe zweier Temperaturfühler, K - T _m Differenz aus mittlerer Heizflächentemperatur und Siedetemperatur, K - T _s örtliches Temperaturgefälle=T _w–T _s, K - T _w Fehler bei der Messung von T_w, K - U _s am Schreiber angezeigte Spannung, V - x Längenkoordinate des Versuchsrohrs, vom unteren Ende des beheizten Teils nach oben gerechnet, cm - z Verstärkungsfaktor des Trennverstärkers - _Cu Wärmeleitfähigkeit des Rohrmaterials, W/cm K - _d Dampfdichte, g/cm³ - _F Flüssigkeitsdichte, g/cm³ Auszug aus der von der Fakultät für Maschinenwesen und Elektrotechnik der Technischen Hochschule München genehmigten Dissertation des Verfassers.     

Power spectra of fluid velocities measured by laser Doppler velocimetry

The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity at each new Doppler burst are studied. Theory valid for low fluctuation intensity flows shows that the measured spectrum is filtered at the mean sample rate and that it contains a filtered white noise spectrum caused by the steps in the sample and hold signal. In the limit of high data density, the step noise vanishes and the sample and hold signal is statistically unbiased for any turbulence intensity.List of symbols A cross-section of the LDV measurement volume, m² - A empirical constant - B bandwidth of velocity spectrum, Hz - C concentration of particles that produce valid signals, number/m³ - d _m diameter of LDV measurement volume, m - f(₁, ₂ | u) probability density of t _i; and t _j given (t) for all t, Hz² - probability density for t _j-t_i, Hz - n (t, t) number of valid bursts in (t, t) = N + n - N (t, t) mean number of valid bursts in (t, t) - N _e mean number of particles in LDV measurement volume - valid signal arrival rate, Hz - mean valid signal arrival rate, Hz - R _uu time delayed autocorrelation of velocity, m²/s² - S _u power spectrum of velocity, m²/s²/Hz - t ₁, t ₂ times at which velocity is correlated, s - t _i, t _j arrival times of the bursts that immediately precede t ₁ and t ₂, respectively, s - t _ij t _j–t _i s - T averaging time for spectral estimator, s - T _u integral time scale of u (t), s - T Taylor's microscale for u (t), s - u velocity vector = U + u, m/s - u fluctuating component of velocity, m/s - U mean velocity, m/s - u _m sampled and held signal, m/s Greek symbols (t) noise signal, m/s - _m (t) sampled and held noise signal, m/s - bandwidth of spectral estimator window, radians/s - time between arrivals in pdf, s - Taylor's microscale of length = UT m - kinematic viscosity - ₁, ₂ arrival times in pdf, s - root mean square of noise signal, m/s - _u root mean square of u, m/s - delay time = t ₂ - t ₁ s - B duration of a Doppler burst, s - circular frequency, radians/s - _c low pass frequency of signal spectrum radians/s Other symbols ensemble average - conditional average - ^ estimate     

Sensitivity of three-segment electrodiffusion probes to eddy shedding

The influence of eddy shedding on the instantaneous readings of a three-segment cylindrical electrodiffusion velocity probe was investigated in an immersed jet with a very low turbulence intensity, = 1.2%. The velocity fluctuations measured by the three-segment probe were smaller than 2.6%, and the maximum error in the flow angle estimation was 2. Vortices with the Strouhal frequency were detected by a simple electrodiffusion probe placed downstream of the three-segment probe, but no peaks with this frequency were found on the frequency spectra of the three-segment probe. From the probe response to a stepwise change of the polarization voltage the characteristic times of the transient process were estimated. List of symbols a parameter in Eq. (1) [A s^b m^-b] - A amplitude gain - b parameter in Eq. (1) - c parameter in Eq. (3) [A s^–1/2] - d probe diameter [m] - f frequency [s^–1] - f _s recording frequency [s^–1] - G power spectrum - I _k relative current through k-th segment, Eq. (2) - i total current [A] - i _k current through k-th segment [A] - N number of data samples - Re Reynolds number, - Sr Strouhal number, - t time [s] - t ₀ characteristic transient time [s] - v jet velocity [m s^-1] - v time mean value of velocity [m s^-1] - v _x, y velocity components measured by probe [m s^-1] - var variance, var - dynamic viscosity [Pa s] - density [kg m^-3] - relative deviation, [%] - flow angle, see Fig. 1 - dimensionless frequency For the financial support of this work we express our thanks to the DFG, Bonn. The assistance of Dr. Ondra Wein and Dr. Pavel Mitschka is greatly appreciated.     

Shear stress and normal stress measurements of aqueous polymer solutions in a cone-and-plate rheogoniometer

Summary The rheological behaviour of aqueous solutions of Separan AP-30 and Polyox WSR-301 in a concentration range of 10–10000 wppm is investigated by means of a cone-and-plate rheogoniometer. The relation between the shear stress and the shear rate is for lower shear rates characterized by a timet ₀, which is concentration dependent. Both polymers show for 4000 s^–1 < < 10000 s^–1 a behaviour similar to that of a Bingham material, characterized by a dynamic viscosity ₀ and an apparent yield stress ₀, which also depend on the concentration. The inertial forces are measured for water and some other Newtonian liquids. An explanation is given why the theoretical model developed for these forces does not match the experimental values; the shape of the liquid surface is shear rate dependent. To obtain the first normal stress difference, we have to correct for these inertial forces, the surface tension and the buoyancy. The normal forces, measured for Separan AP-30, appear to be a linear function of the shear rate for 350 s^–1 < < 3300 s^–1.

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Zusammenfassung Das rheologische Verhalten wäßriger Polymerlösungen von Separan AP-30 und Polyox WSR-301 wird in einem Konzentrationsgebiet von 10–10000 wppm in einem Kegel-Platte-Rheogoniometer untersucht. Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit wird für niedrige Schergeschwindigkeiten durch eine konzentrationsabhängige Zeitt ₀ gekennzeichnet. Für Schergeschwindigkeiten 4000 s^–1 < < 10000 s^–1 zeigen beide Polymere ein genähert binghamsches Verhalten, gekennzeichnet durch eine dynamische Viskosität ₀ und eine scheinbare Fließgrenze ₀, welche ebenfalls konzentrationsabhängig sind. Die Trägheitskräfte werden für Wasser und einige newtonsche Öle bestimmt. Die Abweichung der experimentellen Ergebnisse vom theoretischen Modell wird durch die Abhängigkeit der Gestalt der Flüssigkeitsoberfläche von der Schergeschwindigkeit erklärt. Um die Werte der ersten Normalspannungsdifferenz zu erhalten, muß man bezüglich der Trägheitskräfte, der Oberflächenspannung und der Auftriebskräfte korrigieren. Die Normalspannungen für Separan AP-30, gemessen für 350 s^–1 < < 3300 s^–1, zeigen eine lineare Abhängigkeit von der Schergeschwindigkeit.

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c concentration (wppm) - g acceleration of gravity (ms^–2) - K force (N) - K _b buoyant force (N) - K _c force, acting on the cone (N) - K ₀ dimensional constant def. by eq. [24] (N) - K _s force, def. by eq. [22] (N) - M dimensional constant def. by eq. [24] (Ns) - P _s pressure def. by eq. [17] (Nm^–2) - P ₀ average pressure in the liquid atr = 0 (Nm^–2) - P _R average pressure in the liquid atr = R (Nm^–2) - r ₁,r ₂ radii of curved liquid surface (m) - R platen radius (m) - R _w radius of wetted platen area (m) - S _x standard deviation ofx - t ₀ characteristic time def. by eq. [1] (s) - T temperature (°C) - V volume of the submerged part of the cone (m³) - v tangential velocity of liquid (ms^–1) - x distance (m) - angle (rad) - ₀ cone angle (rad) - calibration constant (Nm^–3) - shear rate (s^–1) - dynamic viscosity (mPa · s) - ₀ viscosity def. by eq. [1] (mPa · s) - contact angle (rad) - density (kgm^–3) - static surface tension (Nm^–1) - shear stress (Nm^–2) - ₀ yield stress def. by eq. [1] (Nm^–2) - _c, _p angular velocity (c = cone,p = plate) (s^–1) With 8 figures and 3 tables     

Consecutive chemical reactions in an annular reactor with non-newtonian flow

The purpose of this paper is to analyze the homogeneous consecutive chemical reactions carried out in an annular reactor with non-Newtonian laminar flow. The fluids are assumed to be characterized by a Ostwald-de Waele (powerlaw) model and the reaction kinetics is considered of general order. Effects of flow pseudoplasticity, dimensionless reaction rate constants, order of reaction kinetics and ratio of inner to outer radii of reactor on the reactor performances are examined in detail.Nomenclature c _A concentration of reactant A, g.mole/cm³ - c _B concentration of reactant B, g.mole/cm³ - c _A0 inlet concentration of reactant A, g.mole/cm³ - C ₁ dimensionless concentration of A, c _A/c _A0 - C ₂ dimensionless concentration of B, c _B/c _A0 - C ₁ dimensionless bulk concentration of A - C ₂ dimensionless bulk concentration of B - D _A molecular diffusivity of A, cm²/sec - D _B molecular diffusivity of B, cm²/sec - k _A first reaction rate constant, (g.mole/cm³)^1–m /sec - k _B second reaction rate constant, (g.mole/cm³)^1–n /sec - K ₁ dimensionless first reaction rate constant, k _A r ₀ ² c _A0 ^m–1 /D _A - K ₂ dimensionless second reaction rate constant, k _B r ₀ ² c _A0 ^n–1 /D _B - K apparent viscosity, dyne(sec) ^m /cm² - m order of reaction kinetics - n order of reaction kinetics - P pressure, dyne/cm² - r radial coordinate, cm - r _i radius of inner tube, cm - r _max radius at maximum velocity, cm - r _o radius of outer tube, cm - R dimensionless radial coordinate, r/r _o - s reciprocal of rheological parameter for power-law model - u local velocity, cm/sec - u _max maximum velocity, cm/sec - u bulk velocity, cm/sec - U dimensionless velocity, u/u - z axial coordinate, cm - Z dimensionless axial coordinate, zD _A/r ₀ ² /u - ratio of molecular diffusivity, D _B/D _A - ratio of inner to outer radius of reactor, r _i/r _o - ratio of radius at maximum velocity to outer radius, r _max/r _o     

Viscous heating and heat transfer in cone-and-plate rheogoniometry

Summary At higher shear rates the relation between shear stress and shear rate appears to deviate from the for Newtonian fluids expected linear behaviour. In cone-and-plate rheogoniometry one of the most important causes of that is the effect of viscous heating. Accurate measurements carried out with a 10 cm diameter cone and plate lead to a semi-logarithmic, linear relationship between temperature increase and time for a Newtonian oil which dynamic viscosity varies approximately linearly with time. A simple model based on a heat balance describes this behaviour quantitatively.

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Zusammenfassung Bei newtonschen Flüssigkeiten weisen die Experimente eine Abweichung vom linearen Zusammenhang zwischen Schubspannung und Schergeschwindigkeit auf. Im Kegel-Platte-Meßsystem ist die Wärmeproduktion durch innere Reibung die wichtigste Ursache der Abweichung. Bei newtonschen Flüssigkeiten, deren dynamische Viskosität sich ungefähr linear mit der Temperatur verändert, ergeben sorgfältig ausgeführte Messungen mit einem Kegel von 10 cm Durchmesser einen linearen Zusammenhang zwischen der Zeit und dem Logarithmus der Temperaturzunahme. Ein aus der Wärmebilanz abgeleitetes Modell vermag dieses Verhalten quantitativ zu beschreiben.

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Symbols A platen surface (m²) - B viscosity constant from eq. [1] (Pa s K^–1) - S _B standard deviation ofB (Pa s K^–1) - S _t0 standard deviation oft ₀ (s) - S _t0 standard deviation oft ₀ (s) - S ₀ standard deviation of ₀ (Pa s) - t time (s) - t ₀ time def. by eq. [5] (s) - t ₀ time def. by eq. [11] (s) - T temperature (°C) - T ₀ temperature of the surrounding air (°C) - T highest experimental temperature (°C) - V volume of the fluid between the platen (m³) - W heat capacity of the system (J K^–1) - heat transfer coefficient (W m^–2 K^–1) - shear rate (s^–1) - dynamic viscosity (Pa s) - ₀ dynamic viscosity atT ₀ (Pa s) - dimensionless temperature def. by eq. [4a] (–) - dimensionless time def. by eq. [4b] (–) - dimensionless time def. by eq. [10] (–) With 4 figures and 2 tables