Mathematical model of a single-screw plasticating extruder

A five zone mathematical model of a plasticating extruder is presented. Its application in the design of new and improvement of existing extruders is briefly described. The model is based on theories proposed by Darnell and Mol, Tadmor, Broyer, McKelvey, Klein, Schneider, Fenner, Poon and Jankov. A comparison between experiments and theoretical calculations is included. E energy, W - f melt film thickness, m - f _k friction coefficient - h channel depth, m - l axial screw distance, m - k power-law parameter, °C^–1 - m ₀ power-law parameter, Pa s ⁿ - MI melt index, g/10 min - n power-law parameter - p pressure, Pa - S screw lead, m - t temperature, °C - t _c time, s - T temperature, K - v velocity, m s^–1 - X solid bed width, m - y rectangular coordinate (channel depth direction), m - Z 1/S (turn), m^–1 - shear rate, s^–1 - apparent viscosity, Pa s - feed angle, ° - density, kg m^–3 - shear stress, Pa - a solid - b barrel or bulk - d dissipated - f flight - m melt - s screw - t total - x width channel direction - z length channel direction     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)     

StationÄrer und instationÄrer Vorgang der Filmkondensation auf der ebenen Platte,dem senkrechten Kegel,dem horizontalen Rohr und der Kugel

Zusammenfassung Die vorliegende Arbeit untersucht die Filmkondensation auf verschiedenen KörperoberflÄchen. Dabei wird sowohl der instationÄre Anlaufvorgang als auch der stationÄre Proze\ betrachtet. Die Ergebnisse für die Schichtdicke des abflie\enden Kondensates werden eingehend diskutiert. Ist die Schichtdicke als Funktion des Ortes und der Zeit bekannt, ist die Berechnung des kondensierenden bzw. abflie\enden Volumenstromes, sowie die Berechnung des lokalen bzw. für die Praxis bedeutungsvolleren globalen WÄrmeübergangs möglich.

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Steady and unsteady process of film condensation on a flat plate, a vertical coin, a horizontal pipe and a sphere

This paper investigates film condensation on different surfaces of geometric bodies. In this connection the unsteady starting process and the steady process are considered. The results for the thickness of layer of the flowing-off condensate are discussed detailed. If the thickness of layer is given as a function of time and location the computation of the condensing, respective flowing-off volume stream and the computation of the local, respective global heat transfer is possible.

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Bezeichnungen C Konstante - R Rohr- bzw. Kugelradius [m] - T Temperatur [K] - kondensierender Volumenstrom pro LÄngeneinheit [m² s^–1] - abflie\ender Volumenstrom pro LÄngeneinheit [m² s^–1] - kondensierender Volumenstrom [m³ s^–1] - abflie\ender Volumenstrom [m³ s^–1] - a Kegelachse - c spez. WÄrme der kondensierenden Flüssigkeit [J kg^–1 K^–1] - e ErzeugendenlÄnge des Kegels, an der die Randbedingung vorgeschrieben ist [m] - g Erdbeschleunigung [m s^–2] - l Platten- bzw. KegellÄnge [m] - p Druck [Nm^–2] - q WÄrmestromdichte [J m^–2 s^–1] - r VerdampfungswÄrme der Flüssigkeit [J kg^–1] - t Zeit [s] - u örtliche Geschwindigkeit des Fluids [m s^–1] - x, y kartesische Ortskoordinaten - r, Zylinder bzw. Kugelkoordinaten - WÄrmeübergangszahl [J m^–2 s^–1] - Neigungswinkel der Platte - öffnungswinkel des Kegels - Schichtdicke der kondensierten Flüssigkeit [m] - WÄrmeleitzahl der kondensierten Flüssigkeit [J m^–1 s^–1] - Dichte der kondensierten Flüssigkeit [kg m^–3] - OberflÄchenspannung der kondensierten Flüssigkeit [Nm^–1] - Schubspannung in der kondensierten Flüssigkeit [Nm^–2] - v kinematische ZÄhigkeit [m² s^–1] - dynamische ZÄhigkeit [kg m^–1 s^–1] - Winkelkoordinate (Rohr, Kugel), bei der eine Randbe-dingung vorgeschieben ist Indizes g gasförmige Phase - m mittlere - s SÄttigungszustand des gasförmigen Mediums - w auf die OberflÄche der Wand (Platte, Kegel, Rohr,Kugel) bezogen - 0 Ursprung der jeweiligen Störungsausbreitung Dimensionslose Kennzahlen Nu Nu\elt-Zahl - Pr Prandtl-Zahl - Re Reynolds-Zahl Kurzfassung der bei Prof. Dr. W. Schneider, Institut für Strömungslehre und WÄrmeübertragung TU Wien, angefertigten Diplomarbeit     

Atmospheric boundary layer simulation in a wind tunnel, using air injection

The inner part of a neutral atmospheric boundary layer has been simulated in a wind tunnel, using air injection through the wind tunnel floor to thicken the boundary layer. The flow over both a rural area and an urban area has been simulated by adapting the roughness of the wind tunnel floor. Due to the thickening of the boundary layer the scaling factor of atmospheric boundary layer simulation with air injection is considerably smaller than that without air injection. This reduction of the scaling factor is very important for the simulation of atmospheric dispersion problems in a wind tunnel.The time-mean velocity distribution, turbulence intensity, Reynolds stress and turbulence spectra have been measured in the inner part of the wind tunnel boundary layer. The results are in rather good agreement with atmospheric measurements.Nomenclature d Zero plane displacement, m - h Height of roughness elements, m - k Von Kármán's constant - n Frequency of turbulence velocity component, s^–1 - S _u(n) Energy spectrum for longitudinal turbulence velocity component, m² s^–1 - S _v(n) Energy spectrum for lateral turbulence velocity component, m² s^–1 - S _w(n) Energy spectrum for vertical turbulence velocity component, m² s^–1 - U _o Free stream velocity outside the boundary layer, m s^–1 - Time-mean velocity inside the boundary layer, m s^–1 - u* Wall-friction velocity, m s^–1 - u Longitudinal turbulence intensity, m s^–1 - v Lateral turbulence intensity, m s^–1 - w Vertical turbulence intensity, m s^–1 - Reynolds stress, m² s^–2 - z Height above earth's surface or wind tunnel floor, m - z _o Roughness length, m - Thickness of inner part of boundary layer, m - Thickness of boundary layer, m - Kinematic viscosity, m² s^–1     

Convective heat transfer in the thermal entrance region of parallel-flow noncircular duct heat exchanger arrays

Convective heat transfer properties of a hydrodynamically fully developed flow, thermally developing flow in a parallel-flow, and noncircular duct heat exchanger passage subject to an insulated boundary condition are analyzed. In fact, due to the complexity of the geometry, this paper investigates in detail heat transfer in a parallel-flow heat exchanger of equilateral-triangular and semicircular ducts. The developing temperature field in each passage in these geometries is obtained seminumerically from solving the energy equation employing the method of lines (MOL). According to this method, the energy equation is reformulated by a system of a first-order differential equation controlling the temperature along each line.Temperature distribution in the thermal entrance region is obtained utilizing sixteen lines or less, in the cross-stream direction of the duct. The grid pattern chosen provides drastic savings in computing time. The representative curves illustrating the isotherms, the variation of the bulk temperature for each passage, and the total Nusselt number with pertinent parameters in the entire thermal entry region are plotted. It is found that the log mean temperature difference (T _LM), the heat exchanger effectiveness, and the number of transfer units (NTU) are 0.247, 0.490, and 1.985 for semicircular ducts, and 0.346, 0.466, and 1.345 for equilateral-triangular ducts.

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Konvektiver Wärmeübergang im thermischen Einlaufgebiet von Gleichstromwärmetauschern mit nichtkreisförmigen Strömungskanälen

Zusammenfassung Die Untersuchung bezieht sich auf das konvektive Wärmeübertragungsverhalten eines Gleichstromwärmetauschers mit nichtkreisförmigen Strömungskanälen bei hydraulisch ausgebildetet, thermisch einlaufender Strömung unter Aufprägung einer adiabaten Randbedingung. Zwei Fälle komplizierter Geometrie, nämlich Kanäle mit gleichseitig dreieckigen und halbkreisförmigen Querschnitten, werden bezüglich des Wärmeübergangsverhaltens bei Gleichstromführung eingehend analysiert. Das sich entwickelnde Temperaturfeld in jedem Kanal von der eben spezifizierten Querschnittsform wird halbnumerisch durch Lösung der Energiegleichung unter Einsatz der Linienmethode (MOL) erhalten. Dieser Methode entsprechend erfolgt eine Umformung der Energiegleichung in ein System von Differentialgleichungen erster Ordnung, welches die Temperaturverteilung auf jeder Linie bestimmt.Die Temperaturverteilung im Einlaufgebiet wird unter Vorgabe von 16 oder weniger Linien über dem Kanalquerschnitt erhalten, wobei die gewählte Gitteranordnung drastische Einsparung an Rechenzeit ergibt. Repräsentative Kurven für das Isothermalfeld, den Verlauf der Mischtemperatur für jeden Kanal und die Gesamt-Nusseltzahl als Funktion relevanter Parameter im gesamten Einlaufgebiet sind in Diagrammform dargestellt. Es zeigt sich, daß die mittlere logarithmische Temperaturdifferenz (T _LM), der Wärmetauscherwirkungsgrad und die Anzahl der Übertragungseinheiten (NTU) folgende Werte annehmen: 0,247, 0,490 und 1,985 für halbkreisförmige Kanäle sowie 0,346, 0,466 und 1,345 für gleichseitig dreieckige Kanäle.

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Nomenclature A cross sectional area [m²] - a characteristic length [m] - C _c specific heat of cold fluid [J kg^–1 K^–1] - C _h specific heat of hot fluid [J kg^–1 K^–1] - C _p specific heat [J kg^–1 K^–1] - C _r specific heat ratio,C _r=C _c/C_h - D _h hydraulic diameter of duct [m] - f friction factor - k thermal conductivity of fluid [Wm^–1 K^–1] - L length of duct [m] - m mass flow rate of fluid [kg s^–1] - N factor defined by Eq. (20) - NTU number of transfer units - Nu _{x, T} local Nusselt number, Eq. (19) - P perimeter [m] - p pressure [KN m^–2] - Pe Peclet number,RePr - Pr Prandtl number,/ - Q _T total heat transfer [W], Eq. (13) - Q ideal heat transfer [W], Eq. (14) - Re Reynolds number,D _h/ - T temperature [K] - T _b bulk temperature [K] - T _e entrance temperature [K] - T _w circumferential duct wall temperature [K] - u, U dimensional and dimensionless velocity of fluid,U=u/u - , dimensional and dimensionless mean velocity of fluid - w generalized dependent variable - X dimensionless axial coordinates,X=D _h ² /a ² x* - x, x* dimensional and dimensionless axial coordinate,x*=x/D _hPe - y, Y dimensional and dimensionless transversal coordinates,Y=y/a - z, Z dimensional and dimensionless transversal coordinates,Z=z/a Greek symbols thermal diffusivity of fluid [m² s^–1] - * right triangular angle, Fig. 2 - independent variable - T _LM log mean temperature difference of heat exchanger - effectiveness of heat exchanger - generalized independent variable - dimensionless temperature - _b dimensionless bulk temperature - dynamic viscosity of fluid [kg m^–1 s^–1] - kinematic viscosity of fluid [m² s^–1] - density of fluid [kg m^–3] - heat transfer efficiency, Eq. (14) - generalized dependent variable     

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)     

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     

A specific slit die rheometer for extruded starchy products. Design,validation and application to maize starch

A novel in-line rheometer, called Rheopac, has been designed and built in order to study the rheological behaviour of starchy products or, more generally, of products sensitive to a thermomechanical treatment. It is based on the principle of a twin channel, using a balance of feed rate between each of them, in order to make local shear rate vary in the measuring section without changing the flow conditions into the extruder. A wide range of shear rate could be reached and measurements were performed more swiftly than with a classical slit die. The viscous behaviour of maize starch was studied by taking into account the influence of the thermomechanical history, which modified the starch degradation and thus led to important variations in the viscosity. Experimental results were satisfactorily compared to previously published models.Nomenclature E activation energy (J · mol^–1) - h channel depth (m) - h ₁ depth under the piston valve in channel 1 (m) - h ₂ depth under the piston valve in channel 2 (m) - K consistency (Pa·s ⁿ ) - K ₀ reference consistency (Pa·s ⁿ ) - L total channel length (m) - L _p length of the piston valve (m) - MC moisture content (wet basis) - n power law index - N screw rotation speed (rpm) - P ₀ entrance pressure (Pa) - P _e pressure at the entry of the piston valve (Pa) - Q ₁ flow rate in channel 1 (m3 · s^–1) - Q ₂ flow rate in channel 2 m³·s^–1) - Q _T total flow rate (m3 · s^–1) - R constant of perfect gas (8.314 J·mol^–1·K^–1) - SME specific mechanical energy (kWh · t^–1) - T temperature (°C) - T _a absolute temperature (K) - T _b barrel temperature (°C) - T _d die temperature (°C) - T _p product temperature (°C) - w channel width (m) - W energetical term (J·m^–3) - viscosity (Pa · s) - [gh ₀] intrinsic viscosity of native starch (ml·g^–1) - [] intrinsic viscosity (ml·g^–1) - shear rate (s^–1) - shear rate in measuring section (s^–1) - maximum shear rate (s^–1)     

Pulsating slurry flow in pipelines

An experimental study on pulsating turbulent flow of sand-water suspension was carried out. The objective was to investigate the effect of pulsating flow parameters, such as, frequency and amplitude on the critical velocity, the pressure drop per unit length of pipeline and hence the energy requirements for hydraulic transportation of a unit mass of solids. The apparatus was constructed as a closed loop of 11.4 m length and 3.3 cm inner diameter of steel tubing. Solid volumetric concentrations of up to 20% were used in turbulent flow at a mean Reynolds number of 33,000–82,000. Pulsation was generated using compressed air in a controlled pulsation unit. Frequencies of 0.1–1.0 Hz and amplitude ratios of up to 30% were used. Instantaneous pressure drop and flow rate curves were digitized to calculate the energy dissipation associated with pulsation. The critical velocity in pulsating flow was found to be less than that for the corresponding steady flow at the same volumetric concentration. Energy dissipation for pulsating flow was found to be a function of both frequency and amplitude of pulsation. A possible energy saving was indicated at frequencies of 0.4–0.8 Hz and moderate amplitudes ratios of less than 25%.List of symbols A cross-section area of the tube (m²) - C _D drag coefficient of sand particles - C _v volumetric concentration (%) - D inner diameter of test-section pipe (m) - F frequency (Hz) - f friction factor - g gravitational constant (m/s²) - J energy dissipation of suspension (W/m)/(kg/s) - J _p energy dissipation of pulsating suspension (W/m)/(kg/s) - J _s energy dissipation of steady component of suspension (W/m)/(kg/s) - J _w energy dissipation of pure water (W/m)/(kg/s) - L length of test-section (m) - m mass flow rate (kg/s) - P pressure drop in test-section (N/m²) - S specific gravity of sand - V instantaneous flow velocity (m/s) - V _c steady flow critical velocity (m/s) - V _cp pulsating flow critical velocity (m/s) - V _F settling velocity of particles (m/s) - V _s steady component of mean flow velocity (m/s) - dynamic viscosity (g/cm sec) - _m mean density of suspension (kg/m³) - angular velocity (rad/sec) - amplitude ratio (V — V _s)/V - nondimentional factor equal to - nondimentional factor equal to (V–V _s/V - NI nondimentional factor equal to (V ²C _d/g D(S – 1)) - Re Reynolds number (V ²C _d/C _v g D(S – 1))     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

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     

Der Einfluß der Diffusion auf die Selektivität der Desorption in einer Rieselfilmsäule

Zusammenfassung Bei der Verdunstung eines Zweistoffgemisches in ein inertes Trägergas in einer Rieselfilmsäule hängt der Trenneffekt nicht allein von der relativen Flüchtigkeit, sondern auch vom Verhältnis der Diffusionsgeschwindigkeiten beider Stoffe im Trägergas ab. Bei der Verdunstung von Isopropanol-Wasser-Gemischen in trockene Luft zeigte sich, daß das Verhältnis der gasseitigen Stoffübergangskoeffizienten bei großen Gasgeschwindigkeiten etwa gleich der Wurzel aus dem Verhältnis der Diffusionskoeffizienten war. Da der Alkolhol im Trägergas langsamer diffundiert als das Wasser, konnten flüssige Mischungen durch absatzweise Verdunstung mit Alkohol angereichert werden, obwohl der Alkohol leichterflüchtig war.Bei kleinen Gasgeschwindigkeiten lieferte der Gleichstrom immer höhere Stoffübergangskoeffizienten als der Gegenstrom. Beim Gleichstrom wurde der Einfluß des Diffusionskoeffizienten auf den Stoffübergangskoeffizienten mit abnehmender Geschwindigkeit größer, beim Gegenstrom wurde er schwächer.

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The influence of diffusion on selectivity of desorption in a wetted wall column

The desorption of a binary mixture into a stripping gas flowing through a wetted-wall column is not only governed by the vapour-liquid-equilibrium. Gas-phase diffusivities of the evaporating components have also to be taken into account. Batch wise stripping experiments of Propanol(2)-water-mixtures using dry air as the stripping gas showed, that at high gas rates the mass transfer coefficients were proportional to the square root of the diffusivities. Therefore it was possible to enrich the residual mixture with Propanol(2) because of its lower diffusivity, although Propanol(2) is more volatile.At low gas rates the mass-transfer coefficients were higher for cocurrent flow than for countercurrent flow. Besides at low gas rates the diffusivities had more influence on mass-transfer for cocurrent flow than for countercurrent flow.

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Abbreviations

Formelzeichen A [m²] Oberfläche des Rieselfilms2 r_ph·L - F [m²] freie Strömungsquerschnittfläche für das Gas in der Rieselfilmsäule: r _ph ² - K _g [–] kinetischer Trennfaktor - k _l [–] Kennzahl für den flüssigseitigen Widerstand - L [m] Länge der Rieselfilmsäule - n [mol/m³] molare Dichte - n _l [mol] Behältermolmenge - N _l,0 [mol] Behältermolmenge zu Beginn des Versuchs - n _i [mol/m² s] Molenstromdichte der Komponentei - N _i [mol/s] Molenstrom der Komponentei - N _g [mol/s] Molenstrom des Trägergases - p [Pa] Druck - p _i ⁰ [Pa] Dampfdruck der reinen Komponente - r [m] Radius - r _i [m] Innenradius des Rieselrohres - r ₁ [–] molarer bezogener Verdunstungsstrom, definiert in Gl. (3) - r ₁ [–] molarer bezogener Verdunstungsstrom, definiert in Gl. (9) - S ₁ [–] Selektivität der Desorption - s _l [m] Filmdicke - u [m/s] Geschwindigkeit - t [s] Zeit - V [m³/s] Volumenstrom - x [–] Molenbruch in der Flüssigkeit - y [–] Molenbruch in der Gasphase - z [m] Längenkoordinate Griechische Buchstaben _T [–] thermodynamischer Trennfaktor - [m/s] Stoffübergangskoeffizient - [–] Aktivitätskoeffizient - [m²/s] Diffusionszahl - [°C] Temperatur - v [m²/s] kinematische Viskosität - [–] Absättigung Indices a Austritt - e Eintritt - g gasseitig - i Komponente - l flüssigseitig - Ph Phasengrenze, Gleichgewicht - RFS Rieselfilmsäule - 1 Isopropanol - 2 Wasser dimensionslose Kennzahlen St _g = _g/¯u _g - Gz _g =4/ V _g/ _g·L - Sh _g = _g·2r _ph - Re _g =¯u _g·2r _ph/v_g - Sc _g =v _g/ _g - NTU _g =·A{itdn_g/N _g - Re _l =V _l/2r _i·v _l     

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     

Flow birefringence of polymer melts: Application to the investigation of time dependent rheological properties

Summary Transient stresses including normal stresses, which are developed in a polymer melt by a suddenly imposed constant rate of shear, are investigated by mechanical measurement and, indirectly, with the aid of the flow birefringence technique. For the latter purpose use is made of the so-called stress-optical law, which is carefully checked.It appears that the essentially linear model of the rubberlike liquid, as proposed byLodge, is capable of describing the behaviour of polymer melts rather well, if the applied total shear does not exceed unity. In order to describe also steady state values of the stresses successfully, one should extend measurements to extremely low shear rates.These statements are verified with the aid of a method which was originally designed bySchwarzl andStruik for the practical calculation of interrelations between linear viscoelastic functions. In the present paper dynamic shear moduli are used as reference functions.

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Zusammenfassung Mit der Zeit anwachsende Spannungen, darunter auch Normalspannungen, wie sie sich nach dem plötzlichen Anlegen einer konstanten Schergeschwindigkeit in einer Polymerschmelze entwickeln, werden mit Hilfe mechanischer Messungen und indirekt mit Hilfe der Strömungsdoppelbrechung untersucht. Für den letzteren Zweck wird das sogenannte spannungsoptische Gesetz herangezogen, dessen Gültigkeit sorgfältig überprüft wird.Es ergibt sich, daß das im Wesen lineare Modell der gummiartigen Flüssigkeit, wie es vonLodge vorgeschlagen wurde, sich recht gut zur Beschreibung des Verhaltens von Polymerschmelzen eignet, solange der im ganzen angelegte Schub den Wert Eins nicht überschreitet. Um auch stationäre Werte der Spannungen in die Beschreibung erfolgreich einzubeziehen, sollte man die Messungen bis zu extrem niedrigen Schergeschwindigkeiten ausdehnen.Die gemachten Feststellungen werden mit Hilfe einer Methode verifiziert, die vonSchwarzl undStruik ursprünglich für die praktische Berechnung von Beziehungen zwischen Zustandsfunktionen entwickelt wurde, die dem linear viskoelastischen Verhalten entsprechen. In der vorliegenden Veröffentlichung dienen die dynamischen Schubmoduln als Bezugsfunktionen.

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a _T shift factor - B _ij Finger deformation tensor - C stress-optical coefficient, (m²/N) - f (p _jl ) undetermined scalar function - G shear modulus, (N/m²) - G(t) time dependent shear modulus, (N/m²) - G() shear storage modulus, (N/m²) - G() shear loss modulus, (N/m²) - G _r reduced shear storage modulus, (N/m²) - G _r reduced shear loss modulus, (N/m²) - H() shear relaxation time spectrum, (N/m²) - k Boltzmann constant, (Nm/°K) - n _ik refractive index tensor - p undetermined hydrostatic pressure, (N/m²) - p _ij ,p _ik stress tensor, (N/m²) - p ₂₁ shear stress, (N/m²) - p ₁₁ –p ₂₂ first normal stress difference, (N/m²) - p ₂₂ –p ₃₃ second normal stress difference, (N/m²) - q shear rate, (s^–1) - t, t time, (s) - T absolute temperature, (°K) - T ₀ reference temperature, (°K) - x the ratiot/ - x position vector of a material point after deformation, (m) - x position vector of a material point before deformation, (m) - ₀, ₁ constants in eq. [37] - ₀, ₁ constants in eq. [37] - shear deformation - (t, t) time dependent shear deformation - _ij unity tensor - n flow birefringence in the 1–2 plane - (q) non-Newtonian shear viscosity, (N s/m²) - ^* () complex dynamic viscosity, (N s/m²) - | ^* ()| absolute value of complex dynamic viscosity, (N s/m²) - () real part of complex dynamic viscosity, (N s/m²) - () imaginary part of complex dynamic viscosity, (N s/m²) - (t — t) memory function, (N/m² · s) - v number of effective chains per unit of volume, (m^–3) - temperature dependent density, (kg/m³) - ₀ density at reference temperatureT ₀, (kg/m³) - relaxation time, (s) - integration variable, (s) - (x) approximate intensity function - ₁ (x) error function - extinction angle - _m orientation angle of the stress ellipsoid - circular frequency, (s^–1) - 1 direction of flow - 2 direction of the velocity gradient - 3 indifferent direction - t time dependence The present investigation has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Research (Z. W. O.).North Atlantic Treaty Organization Science Post Doctoral Fellow.Research Fellow, Delft University of Technology.With 11 figures and 2 tables     

Experimental study of natural convection heat transfer between an outer horizontal cylindrical envelope and an inner concentric heated square cylinder with two slots

This paper presents the results of an experimental study of natural convection heat transfer between a horizontal cylindrical envelope and an internal concentric heated square cylinder with two slots. The internal cylinder was a hollow one with horizontal slots on its top and bottom surfaces. The ratio of slot widthS to the side heightH was 0.0612 and 0.3878. The ratio of the envelope inner diameterD _o to the side heightH was 2.653. Air was used as the working fluid. The range of Ray-leigh number was 1.77×10²8.72×10⁶ forS/H=0.0612 and 1.32×10²6.25×10⁶ forS/H=0.3878. The results show that there are three different heat transfer regimes in different Ray-leigh number regions, i.e. pure conduction regime, transition regime and convection regime. The average heat transfer results were correlated into two empirical equations. Comparison was made with the non-slotted case. It is found that slots of the internal cylinder can significantly enhance the heat transfer.

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Experimentelle Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem konzentrischen, beheizten, quadratischen Prisma mit zwei Schlitzen

Zusammenfassung In der Arbeit werden die Ergebnisse einer experimentellen Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem beheizten quadratischen Prisma mit zwei Schlitzen vorgestellt. Das Prisma selbst ist hohl und weist in der oberen und unteren Begrenzungsfläche je einen horizontalen Längsschlitz auf. Das Verhältnis von SchlitzweiteS zu SeitenhöheH beträgt 0,0612 und 0,3878, das des HülleninnendurchmessersD _o zur SeitenhöheH beträgt 2,653. Als Arbeitsmedium diente Luft. Die Rayleigh-Zahlen variierten zwischen 1,7·10² und 8,72·10⁶ fürS/H=0,0612 und zwischen 1,32·10² und 6,25·10⁶ fürS/H=0,3878. Die Ergebnisse belegen die Existenz dreier unterschiedlicher Wärmeübergangsregime in den verschiedenen Rayleigh-Zahl-Bereichen, und zwar reiner Leitungsbereich, Übergangsgebiet und Konvektionsbereich. Die Ergebnisse für den Wärmeübergang werden im Vergleich mit jenen für ein Prisma ohne Schlitze durch zwei Korrelationbeziehungen dargestellt. Es zeigt sich, daß durch Anbringung von Schlitzen am Innenprisma der Wärmeübergang wesentlich verstärkt werden kann.

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Nomenclature C _p specific heat at constant pressure, J/(kg·K) - D _i diameter of the related circular cylinder whose circumferential area is equal to that of the unslotted square cylinder, m - D _o internal diameter of the outer circular envelope, m - F _i surface area of the inner two slot cylinder, m - g gravitational acceleration, m/s² - H distance between the opposite sides of the square cylinder with two slots, m - K _eq dimensionless equivalent thermal conductivity - L axial length of the test section, m - m ratio of the area of the unslotted square cylinder surface to that of the slotted square cylinder - P pressure in the enclosure, Pa - Q total power input to the enclosure, W - Q _cond radial heat conduction, W - Q _conv convective heat transfer, W - Q _r radiation heat transfer, W - Q _los end heat dissipation, W - R air gas constant, J/(kg·K) - Ra Rayleigh number - S slot width, m - T _i wall temperature of the inner cylinder, K - T _o wall temperature of the outer envelope, K - T _m mean temperature, K - T temperature difference=T _i –T _o , K - W maximum gap width of the test annuli=(D _o –H)/2 for the square case, m Greek symbols ₀ black body radiation constant, W/(m²·K⁴) - _s equation system emissivity - air thermal conductivity, W/(m·K) - _eq equivalent thermal conductivity, W/(m·K) - air dynamic viscosity, kg/(m·s) This work was supported by the National Natural Science Foundation of China.     

Velocity and turbulence measurements in combustion systems

A laser-Doppler velocimeter is used in the measurement of high-temperature gas flows. A two-stage fluidization particle generator provides magnesium oxide particles to serve as optical scattering centers. The one-dimensional dual-beam system is frequency shifted to permit measurements of velocities up to 300 meters per second and turbulence intensities greater than 100 percent.Exiting flows from can-type gas turbine combustors and burners with pre-mixed oxy-acetylene flames are described in terms of the velocity, turbulence intensity, and temperature profiles.The results indicate the influence of the combustion process on turbulence.List of Symbols A exit area of combustor or burner (m²) - A/F mass air-fuel ratio - D exit diameter of combustor or burner (m) - M mass flow rate of gases (kg/s) - N _D number of Doppler bursts used in each velocity measurement - Q volumetric flow rate at T _r (m³/s) - R exit radius of combustor or burner (m) - R _1/2 distance from centerline to radius where the velocity is one-half of the local centerline velocity (m) - Re exit Reynolds number based on cold flow, QD/A - r distance from centerline of flow (m) - T temperature (°C) - T _CL centerline temperature (°C) - T _r inlet (cold) air temperature of combustor or burner (°C) - T.I. turbulence intensity, - mean velocity (m/s) - U _i instantaneous velocity individually realized by LDV (m/s) - mean velocity at centerline of flow (m/s) - mean square velocity fluctuation (m²/s² - x distance along centerline downstream of exit (m) - absolute viscosity at T _r (kg/(ms)) - density at T _r (kg/m³)     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

Messung der Wärmeleitfähigkeit von Stickstoff und Kohlenmonoxid bei hohen Temperaturen im Stoßwellenrohr

Zusammenfassung Die Meßergebnisse für die Wärmeleitfähigkeit von Stickstoff bei Temperaturen zwischen 1230 und 6000 K und Drückenzwischen 1 und 10 bar und von Kohlenmonoxid zwischen 1150 und 5000 K bei 1 bar werden mitgeteilt. Diese mit dem Stoßwellenrohr gemessenen Werte werden mit jenen verglichen, die sich aus der strengen kinetischen Gastheorie ergeben. Auch verfügbare Daten anderer Autoren werden zum Vergleich herangezogen.

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Measurement of thermal conductivity of nitrogen and carbon monoxide at high temperatures in a shock tube

The paper presents results of shock-tube measurements of thermal conductivity of nitrogen at temperatures between 1230 and 6000 K and at pressures between 1 and 10 bar and of carbon monoxide at temperatures between 1150 and 5000 K at 1 bar. Experimental results are compared with several variants of theoretical values, computed from rigorous kinetic theory, and with available data of other authors.

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Bezeichnungen (Einheiten in Klammern) a [m² s^–1] Temperaturleitzahl - C _p[J mol^–1 K^–1] molare Wärmekapazität - k [J K^–1] Boltzmann-Konstante - M [kg mol^–1] molare Masse - p bar Gesamtdruck - R [J mol^–1 K^–1] Gaskonstante - T [K] thermodynamische Temperatur - t [s] Zeit - U [J mol^–1] innere Energie - w [m s^–1] Geschwindigkeit - x [m] Ortskoordinate - x _i [1] Molanteil der Komponentei im Gasgemisch - [Wm^–1 K^–1] Wärmeleitfähigkeit - [mol m^–3] molare Konzentration Indizes i die Komponentei im Gasgemisch - g bezieht sich auf das (kalte) Gas bei der Wandtemperatur - w bezieht sich auf die feste Wand - p bei konstantem Druck Dieser Beitrag wurde auf dem Thermodynamik-Kolloquium des VDI im Oktober 1969 in Zürich vorgetragen.     

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