Heat transfer during dry spinning of fibers

Summary Heat transfer to fibers formed in dry spinning has been subjected to fundamental analysis. Solutions of the equation of energy have been derived and tested with experimental data. Results were deemed satisfactory in view of the accuracy of the experimental data. The present work is believed to yield a good representation of the heat transfer in the dry spinning process.Nomenclature h heat transfer coefficient, cal/cm² °C sec - k thermal conductivity, cal/cm °C sec - r radial distance, cm - t time, sec - Z axial distance, cm - A surface area, cm² - A _n or n-th root of - A ₀ energy required for solvent evaporation, cal/sec cm³ - C _p specific heat cal/gm °C - J ₀ Bessel function of first kind, order zero - J ₁ Bessel function of first kind, order one - J ₂ Bessel function of first kind, order two - N k/(R ² C _p V) - Q volumetric flow rate cm³/sec - R outside radius, cm - T point temperature, °C - T _S surface temperature °C - T ₀ initial fiber temperature °C - T ambient air temperature, °C - average fiber temperature, °C - ₁ average fiber temperature of preceding segment, °C - V average fiber velocity relative to air strean, cm/sec - V _r radial velocity component, cm/sec - V _z axial velocity components, cm/sec - V direction velocity component cm/sec - W weight of solvent evaporated in a given fiber segment, gms - _n a solution of the equation J ₀(X)=0 - heat of vaporization of solvent, cal/gm - dimension - r/R - density, gms/cc     

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     

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     

Swirling turbulent flow through a curved pipe

An experimental study of a swirling turbulent flow through a curved pipe with a pipe-to-mean-bend radius ratio of 0.077 and a flow Reynolds number based on pipe diameter and mean bulk velocity of 50,000 has been carried out. A rotating section, six pipe diameters long, is set up at six diameters upstream of the curved bend entrance. The rotating section is designed to provide a solid-body rotation to the flow. At the entrance of the rotating section, a fully-developed turbulent pipe flow is established. This study reports on the flow characteristics for the case where the swirl number, defined as the ratio of the pipe circumferential velocity to mean bulk velocity, is one. Wall static pressures, mean velocities, Reynolds stresses and wall shear distribution around the pipe are measured using pressure transducers, rotating-wires and surface hot-film gauges. The measurements are used to analyze the competing effects of swirl and bend curvature on curved-pipe flows, particularly their influence on the secondary flow pattern in the crossstream plane of the curved pipe. At this swirl number, all measured data indicate that, besides the decaying combined free and forced vortex, there are no secondary cells present in the cross-stream plane of the curved pipe. Consequently, the flow displays characteristics of axial symmetry and the turbulent normal stress distributions are more uniform across the pipe compared to fully-developed pipe flows.List of symbols B calibration constant - e bridge voltage - e ₀ bridge voltage at zero flow - C _f total skin friction coefficient, = 2 _w/ W ₀ ² - D pipe diameter, = 7.62 cm - De Dean number, = ^1/2 Re - M angular momentum - n calibration constant - N _s swirl number, = D/2 W ₀ - r radial coordinate - R mean bend radius of curvature, = 49.5 cm - Re pipe Reynolds number, = DW ₀/ - S axial coordinate along the upstream (measured negative) and downstream (measured positive) tangent - U, V, W mean velocities along the radial, tangential and axial directions, respectively - u, v, w mean fluctuating velocities along the radial, tangential and axial directions, respectively - u, v, w root mean square normal stress along the radial, tangential and axial directions, respectively - v ^{ov2}, u^{ov2} normal stress along the tangential and radial direction, respectively - W ₀ mean bulk velocity, 10 m/s - W _c W measured at pipe axis - W total wall friction velocity, - total wall friction velocity measured at S/D = -18 - ,v vw, w7#x016B; turbulent shear stresses - pipe-to-mean-bend radius ratio, = D/2 R = 0.077 - axial coordinate measured from bend entrance - fluid kinematic viscosity - fluid density - _w mean total wall shear stress - instantaneous total wall shear - azimuthal coordinate measured zero from pipe hori zontal diameter near outer bend - angular speed of the rotating section     

Viscoelastic swirling flow with free surface in cylindrical chambers

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

An analytical approach to coupled heat and moisture transport in soil

The simultaneous diffusion of heat and moisture through soil is described by two coupled partial differential equations in which the diffusion coefficients are highly non-linear functions of the dependent variables. The system has been regarded as analytically intractable for any generality of coupled flow. However, for an asymptotically steady state, the equations show a marked periodic stability. Computer simulation indicates that the behaviour quickly becomes entrained to input boundary periodicity for any initial state, regardless of the detailed functional form of the diffusion coefficients. This property allows an harmonic series solution to be assembled. Factors such as amplitude decay, phase shift and wave form evolution may be evaluated. The solution is adapted to boundary conditions pertaining to arid soils and the results validated against the 1968 field data of Rose and the 1973 experiment by Jackson.Notation gradient operator - divergence operator - _A amplitude of surface moisture content variation - _l volumetric liquid content, m³/m³ - _c value for moisture content, at which vapour diffusivity decays to zero - _M mean of surface moisture content variation - _s saturation value of moisture content - tortuosity factor, m/m - _i eigenvalues of ₀ - hypothetical thermal conductivity, J/m/sec/K - ₀ density of saturated water vapour, kg/m³ - _l density of liquid water, kg/m³ - _v density of water vapour, kg/m³ - surface tension, kg/sec² - matric potential, m - C volumetric heat capacity, J/m³/K - D ^* molecular diffusivity of water vapour in the porous medium, m²/sec - D _atm molecular diffusivity of water vapour in air, m²/sec - D _TV thermally induced vapour diffusivity, m²/sec/K - D _Tl thermally induced liquid diffusivity, m²/sec/K - D _v isothermal vapour diffusivity, m²/sec - D _l isothermal liquid diffusivity, m²/sec - L latent heat of vaporisation, J/kg - P atmospheric pressure at soil surface,Pa - R gas constant of water vapour, J/kg/K - T temperature,K - T _M mean temperature at surface, K - T _A temperature amplitude at surface, K - g acceleration due to gravity, m/sec² - h relative humidity, dimensionless - p partial pressure of water vapour,Pa - q _v water vapour flux, kg/m²/sec - t time, sec - z depth, (measured downwards), m     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Die Temperaturabhängigkeit der Oberflächenspannung von reinen Kältemitteln vom Tripelpunkt bis zum kritischen Punkt

Zusammenfassung Die Oberflächenspannung von sechs reinen Substanzen — SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃ und CHClF₂ — wurde mit Hilfe einer modifizierten Kapillarmethode gemessen. Die zur Berechnung der Oberflächenspannung erforderlichen Sättigungsdichten und wurden teils aus vorhandenen Zustandsgleichungen, teils aus ebenfalls gemessenen Brechungsindizes bestimmt. Die Temperaturabhängigkeit der Oberflächenspannung läßt sich durch einen erweiterten Ansatz nach van der Waals =_O (T_c-T)(1+...) darstellen, wobei bei einfachen Stoffen ein eingliedriger, bei polaren und assoziierenden Stoffen ein zweigliedriger Ansatz notwendig und ausreichend ist. Für den kritischen Exponenten der Oberflächenspannung wurde ein von der molekularen Substanz weitgehend unabhängiger Wert von =1.284±0.005 gefunden.

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Temperature dependence of surface tension of pure refrigerants from triple point up to the critical point

The surface tension of six fluids (SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃, CHClF₂) have been measured by means of a modified capillary rise method. The liquid vapor densities, which are needed to calculate the surface tension, have partly been determined by means of refractive indices simultaneously measured in the same apparatus. The temperature dependence of the surface tension is described by an extended van der Waals power law =_O(T_c-T)(1+...). For simple fluids one term and for polar and associating fluids two terms are necessary and sufficient. The critical exponent is found to be 1.284 ± 0.005 and nearly independent of the molecular structure.

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Formelzeichen a² Laplace-Koeffizient - a Parameter - B_O, B_on Koeffizient der Koexistenzkurve - g Erdbeschleunigung - H Höhe, kapillare Steighöhe - LL Lorentz-Lorenz-Funktion oder Refraktionskonstante - M molare Masse - M Zahl der Meßwerte - N Zahl der unbekannten Parameter - n Brechungsindex - p Druck - R,r Radius - s Entropie - SD Standardabweichung - T, t Temperatur - u innere Energie Griechische Formelzeichen Exponent des Laplace-Koeffizienten - Exponent der Koexistenzkurve - 2. Exponent der Oberflächenspannung - Wellenlänge des Lichts - Exponent der Oberflächenspannung - _D Dipolmoment - , Dichte der Flüssigkeit bzw. des Dampfes - Oberflächenspannung - reduzierte Temperatur (1-T/T_c) - ² gewichtete Varianz Indizes c kritischer Zustand - D Differenz - m Mittelwert - T Isotherme - t Zustand am Tripelpunkt - S Zustand am Schmelzpunkt - bezogen auf Oberfläche     

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     

Experimental investigation of a hybrid wind tunnel model

New developments in power electronics (such as the linear motor) may be of great advantage for future economic aerodynamic testing. After a comprehensive review of economical aspects of present wind tunnel technologies, the experimental verification of the hybrid wind tunnel concept will be presented. A scale law is derived to extrapolate the experimental results of the model-HWT to a facility size for simulating full scale Re-numbers. Finally, methods will be discussed by which the flow constancy during the running time may be improved.List of symbols A area - a velocity of sound - d diameter - i influence region - l length - Ma Mach number - m facility mass - n exponent in power law of boundary layer profile u/u ₂ = (y/)^1/n - P power - p pressure - q kinetic pressure (=(/2)u ²) - Re Reynolds number - r radius - t time - u, w velocity - V volume - x, y coordinates - boundary layer thickness - ^* boundary layer displacement thickness - velocity deficit at boundary layer edge (usually defined as = 0.01) - windtunnel efficiency factor - efficiency - ratio of specific heats - test facility mass ratio - kinematic viscosity - normalized coordinate - mass density - stress - ( )_1,2,3 flow region (Fig. 1) presuffix 0 signifies - ( )_2,2,2 flow state (Figs. 7; 11) stagnation condition - ( ) test section values - ( ) _c cryo-windtunnel - ( ) _d diffusor - ( ) _D model drag - ( ) _el electric - ( ) _ew1 head of expansion wave - ( ) _ew2 tail of expansion wave - ( ) _f fan - ( ) _fs fan shaft - ( ) _mt model tunnel - ( ) _N normal temperature windtunnel - ( ) _or original tunnel - ( ) _r,l right, leftrunning waves - ( ) _rp wind tunnel return pipe - ( ) _s suction tube - ( ) _sl test sled - ( ) _sw shock wave - ~ relative system - - values normalized with a ₁ or p ₁ - ETW European Transonic Windtunnel (early, noncryogenic concepts. A = 21 m₂) - HWT Hybrid-Wind-Tunnel - LIM Linear Induction Motor - LN₂ Liquid Nitrogen - LT Ludwieg-tube     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system

Summary Starting with an assumed relationship between the stress tensor, the non-Newtonian viscosity, and the strain rate tensor, the nonlinear equations of motion are developed for use in any orthogonal coordinate system. The resulting equations are written in terms of the scalar velocities, the non-Newtonian viscosity, the metric coefficients, and their derivatives.The non-Newtonian viscosity is assumed to be a scalar function of the strain rate tensor, and so depends upon the invariants of the strain rate tensor. For convenience, the necessary invariants are written out in complete form for use in any orthogonal coordinate system, in terms of the scalar velocities, the metric coefficients, and their derivatives.Using the resulting motion equations and a model of this type of viscosity, theOstwald-de Waele model, an example of time dependent flow is solved using a continuous time-discrete space method programmed on an analog computer. e _ij strain rate tensor - body force density, dynes/cm³ - F ₁,F ₂,F ₃ components of body force density, dynes/cm³ - g acceleration of gravity - H function of time - h ₁,h ₂,h ₃ metric coefficients - I ₁,I ₂,I ₃ invariants - m constant - P pressure, dynes/cm² - r radius, cm - t time, sec - velocity vector, cm/sec - v ₁,v ₂,v ₃ velocities in thex ₁,x ₂ andx ₃ directions, respectively, cm/sec - v _n (t) velocity of thenth node, cm/sec - x ₁,x ₂,x ₃ coordinate directions - z coordinate, cm - unit tensor - _ij Kronecker delta - _ij 2e _ij - nabla - _ijk alternating unit tensor - non-Newtonian viscosity, dynes/cm² - ₀, ₁ constant viscosities, dynes sec/cm², dynes sec ^m /cm² - angle, radians - v ₀,v ₁ constant kinematic viscosities, cm²/sec, cm² sec ^m-2 - density, g/cm³ - _ij stress tensor - fluid dilation With 3 figures     

On the measurement of normal stress differences in steady shear flow

Summary Compared to the similar pressure-distribution cone-and-plate apparatus of Adams and Lodge (4), the new apparatus' improvements include: temperature control of the cone (as well as the plate); increased stiffening of the frame; four (not three) pressuremeasuring holes in the cone/plate region; inclusion of a pressure-measuring hole on the axis under the cone truncation; exclusive use of a vertical free liquid boundary at the cone rim (without a sea of liquid). Temperature control of the rotating cone and of the fixed plate leads to acceptable temperature uniformity in the test liquid for working temperatures within 10°C or 20°C of ambient; the corresponding interval is about 1°C if the cone temperature control is abandoned. Pressure gradients measured using a Newtonian liquid agree with those calculated using Walters' eq. (3). For a viscoelastic liquid, after subtracting inertial contributions, pressure distributions measured at a given shear rate in the cone/plate region do not change when the gap angle is changed from 2° to 3°, showing that the effects of secondary flow are negligible. Values ofN ₃ =N ₁ + 2N ₂ obtained from the gradients of these distributions are believed to be in error by not more than ±1 Pa, in favorable cases. The present most useful ranges are: 10 to 5000 Pa forN ₃; 0.1 to 200 sec^–1 for shear rate; up to 5 Pa s for viscosity; and 5 to 40°C for temperature. As an application, it is shown that adding 0.1% of a high molecular weight polyisobutylene to a 2% polyisobutylene solution doublesN ₃ and has no detectable effect on the viscosity measured at low shear rates with a Ferranti-Shirley viscometer.

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Zusammenfassung Im Vergleich zu dem ähnlichen Kegel-Platte-Gerät von Adams und Lodge (4) zur Messung der Druckverteilung wurden an dem neuen Gerät die folgenden Verbesserungen vorgenommen: Temperaturregelung an Kegel und Platte, Versteifung des Rahmens, vier (anstatt drei) Druckmeßlöcher im Kegel-Platte-Bereich, ein zusätzliches Druckmeßloch auf der Achse unter der Kegelstumpf-Deckfläche, ausschließliche Verwendung einer vertikalen freien Grenzfläche der Flüssigkeit am Kegelrand (ohne umgebenden Flüssigkeitssee). Die Temperaturregelung des rotierenden Kegels und der festen Platte führt zu einer ausreichenden Temperaturgleichförmigkeit in der Testflüssigkeit für Betriebstemperaturen, die höchstens um 10–20°C von der Umgebungstemperatur abweichen. Dieses Intervall beträgt dagegen nur etwa 1°C, wenn auf die Temperaturregelung am Kegel verzichtet wird. Für newtonsche Flüssigkeiten entsprechen die gemessenen Druckgradienten den mittels der Gleichung von Walters (3) berechneten. Für viskoelastische Flüssigkeiten zeigen sich bei der Änderung des Spaltwinkels von 2° auf 3° nach Abzug der Trägheitsbeiträge keine Änderungen der bei einer bestimmten Schergeschwindigkeit gemessenen Druckverteilung. Dies zeigt, daß Sekundärströmungseffekte vernachlässigbar sind. Es darf angenommen werden, daß die Werte vonN ₃ =N ₁ + 2N ₂, die man aus den Gradienten dieser Verteilungen erhält, unter günstigen Umständen mit einem Fehler von nicht mehr als ±1 Pa behaftet sind. Gegenwärtig liegen die günstigsten Bereiche bei 10 bis 5000 Pa fürN ₃, 0,1 bis 200 s^–1 für die Schergeschwindigkeit, unterhalb von 5 Pa s für die Viskosität und 5 bis 40°C für die Temperatur. Als Anwendung wird gezeigt, daß ein Zusatz von 0,1% hochmolekularen Polyisobutylens zu einer 2%igen Polyisobutylenlösung den Wert vonN ₃ verdoppelt, aber keinen erkennbaren Einfluß auf die (bei geringen Schergeschwindigkeiten mit einem Ferranti-Shirley-Viskosimeter gemessen) Viskosität hat.

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udsf unidirectional shear flow - TCP truncated-cone and plate - N ₁,N ₂ 1st and 2nd normal stress differences in udsf - N ₃ N ₁ + 2N ₂ - : = A is defined by the equationA := B - P ^* hole pressurePw – Pm; Pw, Pm = pressures measured by flush transducer and by hole-mounted transducer - t time - , strain rate, shear rate - (P,t) covariant body metric tensor at particleP and timet - ⁱ , _i covariant and contravariant udsf body base vectors (i = 1, 2, 3) - ^–1 inverse of - R, plate radius, cone/plate gap angle - r ₀,h ₀ radius and height of cone truncation - r,, spherical polar coordinates; cone axis = 0; plate surface = /2 - physical components of stress; for a tensile component - cone angular velocity - p on the plate = /2 - ,T, density, absolute temperature, viscosity - P 0.15 ²(r ² –R ²) (inertial contribution) [2.7] - P _ve contribution [2.8] from flow perturbations of viscoelastic origin - r _i i = 1,2,3,4; values ofr at centers of holes in cone/plate region - P _i () pressure change recorded by transducerTi when cone angular velocity goes from zero to - 1/2 {P _i ()+ P _i (–)} (average for 2 senses of rotation) - rim pressure, from least-squares line through four points - Re Reynolds' number:R ²/ - (P,t)/t With 11 figures and 2 tables     

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     

Berechnung des dynamischen Verhaltens von instationären Temperaturfeldern in zylindrischen Bauteilen

Zusammenfassung Im ersten Teil dieser Untersuchung wird zur Betrachtung des dynamischen Verhaltens instationärer Temperaturfelder in den Wandungen zylindrischer Rohre ein mathematisches Modell erstellt und mit Hilfe der Laplace-Transformation ausgewertet. Im einzelnen werden dabei die Übertragungsfunktionen der Rohrwandtemperaturen hergeleitet und für den Fall der Abweichung vom stationären Zustand unter dem Einfluß äußerer Störungen explizit dargestellt.Im zweiten Teil der Untersuchung wird das sich daraus ergebende dynamische Verhalten der Wandtemperatur fluiddurchströmter Rohre für einige Beispiele in Form von Ortskurven dargestellt.

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Computation of the dynamic behaviour of unsteady-state temperature fields in cylindrical structures

In the first part of this paper a mathematical model is developed allowing the investigation of unsteady-state temperature fields in the walls of cylindrical pipes. Evaluation is done by means of Laplace-transformation. In particular the transfer function of the pipe wall temperature is derived, explicitly shown for the case of deviations from steady-state influenced by external disturbances. In the second part of this paper the resulting dynamical behaviour of the wall temperature of heat pipes containing a fluid is shown by means of Nyquist plots for several examples.

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Formelzeichen a Temperaturleitzahl m²/sec - A, B, A^*, B^*, , ¯B Integrationskonstanten °C - ber, bei, ker, kei Kelvin-Funktionen - ber₁, bei₁, ker₁, kei₁ kelvin-Funklionerion - Bi Biot-Zahl - c spezifische Wärme kJ/kg K - F Übertragungsfunktion - i –1 (imaginäre Einheit) - I₀, K₀, I₁, K₁ modifizierte Bessel-Funktionen - N Nenner (Gl. (39)) - r Rohrradius m - R normierter Abstand von der Innenwand % - s (komplexe) Laplace-Variable 1/sec - t Zeitvariable sec - T Zeitkonstante sec - u Integrationsvariable (Gl. (15)) - Y₀₀, Y₁₀, Y₁₁ Hilfsfunktionen (Gl. (35)-(37)) - Wärmeübergangszahl kW/K m² - kleine Änderung - Laplace-Operator 1/m² - Umgebungstemperatur °C - Rohrwandtemperatur °C - Wärmeleitfähigkeit kW/K m - Dichte kg/m³ - (komplexe) Kennvariable (Gl. (11)) - Frequenz 1/sec - Variable (Gl. (45)) Indizes a Rohraußenwand - FDS Frischdampfsammelrohr - F Fluid - H Heizgas - i Rohrinnenwand - m Mittel - VD Verdampferrohr - W Rohrwand - 0 zum Zeitpunkt t=t₀ - -(Überstreichung) stationärer Zustand Herrn Prof. Dr.-Ing. R. Quack zum 65. Geburtstag gewidmet.     

Flow in porous media I: A theoretical derivation of Darcy's law

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.The presence of spatial deviations of the pressure and velocity in the volume-averaged equations of motion gives rise to aclosure problem, and representations for the spatial deviations are derived that lead to Darcy's law. The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem ofabrupt changes in the structure of a porous medium is not considered.Roman Letters A interfacial area of the - interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the - interface contained within the averaging volume, m² - A ^* interfacial area of the - interface contained within a unit cell, m² - Ae area of entrances and exits for the -phase contained within a unit cell, m² - B second order tensor used to represent the velocity deviation (see Equation (3.30)) - b vector used to represent the pressure deviation (see Equation (3.31)), m^–1 - d distance between two points at which the pressure is measured, m - g gravity vector, m/s² - K Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the -phase (see Figure 2), m - characteristic length scale for the -phase (see Figure 2), m - n unit normal vector pointing from the -phase toward the -phase (n =–n ) - n _e unit normal vector for the entrances and exits of the -phase contained within a unit cell - p pressure in the -phase, N/m² - p intrinsic phase average pressure for the -phase, N/m² - p –p , spatial deviation of the pressure in the -phase, N/m² - r ₀ radius of the averaging volume and radius of a capillary tube, m - v velocity vector for the -phase, m/s - v phase average velocity vector for the -phase, m/s - v intrinsic phase average velocity vector for the -phase, m/s - v –v , spatial deviation of the velocity vector for the -phase, m/s - V averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ Greek Letters V/V, volume fraction of the -phase - mass density of the -phase, kg/m³ - viscosity of the -phase, Nt/m² - arbitrary function used in the representation of the velocity deviation (see Equations (3.11) and (B1)), m/s - arbitrary function used in the representation of the pressure deviation (see Equations (3.12) and (B2)), s^–1     

Shock heating of plasma in a rapidly increasing magnetic field

The experimental excitation of intense collisionless shock waves (M 5) with subsequent plasma compression by the magnetic field of a shock coil is described. A magnetic plug > 20 kOe is produced in 100 × 10^–9 sec by a current generator, a long line with 250-kV water insulation and a characteristic impedance of l At an initial deuterium-plasma density of 2 × 10¹⁴ cm^–3, shock waves with a front width of 20c/₀₃and a velocity of 5 × 10₇ cm/sec are recorded. The ion energy after the accumulation, determined from the neutron yield, turns out to be 2 ke V. Axial shock waves excited by the plasma flow beneath the shock coil are observed.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, Vol. 11, No. 2, pp. 28–38, March–April, 1970.The authors thank G. I. Budker and R. Z. Sagdeev for formulating the problem, R. I. Soloukhin for interest in the study, and S. P. Shalamov for construction of the apparatus.     

Stability characteristics of condensate films

The linear stability theory is used to study stability characteristics of laminar condensate film flow down an arbitrarily inclined wall. A critical Reynolds number exists above which disturbances will be amplified. The magnitude of the critical Reynolds number is in all practical situations so small that a laminar gravity-induced condensate film can be expected to be unstable. Several stabilizing effects are acting on the film flow; at an inclined wall these effects are due to surface tension, gravity and condensation mass transfer.

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Zusammenfassung Mit Hilfe der linearen Stabilitätstheorie werden die Stabilitätseigenschaften laminarer Kondensatfilme an einer geneigten Wand untersucht. Es zeigt sich, daß Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einfluß von Oberflächenspannung, Schwerkraft und Stoffübertragung durch Kondensation bewkkt jedoch, daß Störungen in bestimmten Wellenlängenbereichen gedämpft werden.

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Nomenclature c=c^*/u₀ complex wave velocity, celerity, dimensionless - c^*=c _r ^* + i c _i ^* complex wave velocity, celerity, dimensional - c_p specific heat at constant pressure - g gravitational acceleration - h_fg latent heat - k thermal conductivity of liquid - p^* pressure - p=p^*/u₀ ² dimensionless pressure - Pe=Pr Re^* Peclet number - Pr Prandtl number - Re^*=u₀ / Reynolds number (defined with surface velocity) - S temperature perturbation amplitude - t^* time - t=t^* u₀/ dimensionless time - T temperature - T_s saturation temperature - T_w wall temperature - T=T_s-T_w temperature drop across liquid film - u^*, v^* velocity components - u=u^*/u₀ dimensionless velocity components - v=v^*/u₀ dimensionless velocity components - u₀ surface velocity of undisturbed film flow - v _g ^* vapor velocity - x^*, y^* coordinates - x=x^*/ dimensionless coordinates - y=y^*/ dimensionless coordinates Greek Symbols =^* wave number, dimensionless - ^*=2 /^* wave number dimensional - ^* wave length, dimensional - =^*/ wave length, dimensionless - local thickness of undisturbed condensate film - kinematic viscosity, liquid - density, liquid - _g density vapor - surface tension - = (1 +) film thickness of disturbed film, Fig. 1 - stream function perturbation amplitude - angle of inclination Base flow quantities are denoted by^–, disturbance quantities are denoted by.     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress