Study of the non-isothermal glass fibre drawing process

The glass fibre drawing process is simulated using a finite-element method. The two-dimensional energy and momentum equations are solved in their fully non-linear forms. These are coupled via the temperature-sensitive viscosity function. Both convective and radiative cooling mechanisms are taken into account on the filament surface. An effective emissivity of about 0.2 is found to be applicable to the drawing conditions in this paper. Even at this fairly low effective emissivity, radiation is found to be the dominant mode of cooling. The material thermal conductivity is found to have a small but definite influence on the filament profiles. Two-dimensionsl effects of the kinematic field are only significant up to a distance of about two orifice radii from the nozzle exit.The symbols in the square brackets show the dimensions of the parameters;M Mass,L Length,T Temperature,t Time. a Constant radius of a uniform cylinder [L] - A Local cross-sectional area of the filament [L ² ] - b _i Total tension applied on the filament boundary surface in thei ^th direction [ML/t ² ] - c Specific heat [L ² /t ² T] - D Local filament diameter [L] - f _i i ^th component of the body-force vector [L/t ² ] - h Surface convective heat transfer coefficient of the filament [M/t ³ T] - H Total equivalent heat transfer coefficient due to both convection and radiation [M/t ³ T] - k Thermal conductivity [ML/t ³ T] - M Mass-flow rate [M/t] - n Coordinate normal to the local filament surface [L] - Nu Local Nusselt number [–] - Average Nusselt number [–] - Q Rate of heat transfer [ML ² /t ³ ] - Volume-flow rate [ ³ /t] - r Radial coordinate [L] - R Local radius of the filament [L] - Re _x Reynolds number based on characteristic length scalex [–] - s Coordinate along the filament surface [L] - T Temperature [T] - u Radial component of the velocity [T/t] - U Free-stream velocity of a uniform flow [L/t] - v Local speed of a fluid particle defined by v = ;[L/t] - V Volume [L ³ ] - v _f Constant velocity of a filament with a uniform radius [L/t] - w Axial component of the velocity [L/t] - Average axial velocity of the fluid inside the tube [L/t] - z Axial coordinate, i.e. axial distance from the orifice exit [L] - Exponential coefficient of the viscosity function [T ^–1 ] - _ij Kronecker delta [–] - Emissivity or total hemispherical emissivity [–] - µ Viscosity [M/Lt] - µ ₀ Reference viscosity defined byµ = µ ₀ e ^–T [M/Lt] - Fluid density [M/L ³ ] - Stefan-Boltzmann constant [M/t ³ T ⁴ ] - Viscous dissipation function [M/Lt ³ ] - a Of air - a Based on the (constant) filament radius - C.L. Referred to the centre line of the filament - conv Referred to convection - D Dased on the diameter - f Referred to the filament local condition - g Referred to glass - i,j Species in multi-component systems - o Quantity evaluated at the orifice exit - R Based on the radius - rad Referred to radiation - s Evaluated at the filament surface - tot Referred to the total heat transfer from the filament surface - w Evaluated at the tube wall - Ambient condition - * Refers to non-dimensional quantities - — Indicating quantities averaged over the filament cross-section     

Diffusion of moisture in deformable porous media. Anisotropic fibrous materials

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

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     

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     

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     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

Sensitivity of three-segment electrodiffusion probes to eddy shedding

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

Self-similar solution for deep-penetrating hydraulic fracture propagation

The propagation of a vertical hydraulic fracture of a constant height driven by a viscous fluid injected into a crack under constant pressure, is considered. The fracture is assumed to be rectangular, symmetric with respect to the well, and highly elongated in the horizontal direction (the Perkins and Kern model). The fracturing fluid viscosity is assumed to be different from the stratum saturating fluid viscosity, and the stratum fluid displacement by a fracturing fluid in a porous medium is assumed to be piston-like. The compressibility of the fracturing fluid is neglected. The stratum fluid motion is governed by the equation of transient seepage flow through a porous medium.A self-similar solution to the problem is constructed under the assumption of the quasi-steady character of the fracturing fluid flow in a crack and in a stratum and of a locally one-dimensional character of fluid-loss through the crack surfaces. Crack propagation under a constant injection pressure is characterized by a variation of the crack sizel in timet according to the lawl(t)=l _o (1+At)^1/4, where the constantA is the eigenvalue of the problem. In this case, the crack volume isVl, the seepage volume of fracturing fluidV _f l ³, and the flow rate of a fluid injected into a crack isQ ₀l ^–1.     

Analytical approximation for nonlinear adsorbing solute transport and first-order degradation

Under some constraints, solutes undergoing nonlinear adsorption migrate according to a traveling wave. Analytical traveling wave solutions were used to obtain an approximation for the solute front shape,c(z, t), for the situation of equilibrium nonlinear adsorption and first-order degradation. This approximation describes numerically obtained fronts and breakthrough curves well. It is shown to describe fronts more accurately than a solution based on linearized adsorption. The latter solution accounts neither for the relatively steep downstream solute front nor for the deceleration in time of the nonlinear front.Notation A parameter - c concentration [mol/m³] - c ₀ ^* depth-dependent local maximum concentration [mol/m³] - c; c ₀;c _i concentration difference, feed, and initial resident concentrations, respectively [mol/m³] - D pore scale diffusion/dispersion coefficient [m²/yr] - f adsorption isotherm - f derivative off toc - f second derivative off toc - G ^* parameter - K nonlinear adsorption coefficient [mol/m³)^1–n ] - l column length [m] - L _d dispersivity [m] - m parameter - n Freundlich sorption parameter - P function ofc ₀ ^* - _q change inq [mol/m³] - q adsorbed amount (volumetric basis) [mol/m³] - q derivative ofq toc - R nonlinear retardation factor - retardation factor for concentrationc - R _l linear retardation factor - R(z ^*) depth-dependent average retardation factor, for front at depthz ^* - s adsorbed amount (mass basis) [mol/kg] - t time [years] - u parameter - v flow velocity [m] - z ^* downstream front depth [m] - z depth [m] - transformed coordinate [m] - ^* reference point value of [m] - first-order decay parameter [y^–1] - dry bulk density [kg/m³] - volumetric water fraction - parameter     

Strömungsmodell der Gas-Flüssigreaktoren nach der Filmtheorie

Zusammenfassung Es wird ein mathematisches Strömungsmodell für Gas-Flüssigreaktoren aufgestellt, das auf der Filmtheorie basiert. Für den Fall einer chemischen Reaktion erster Ordnung läßt sich eine geschlossene analytische Lösung finden, mit deren Hilfe man den Stoffaustauschgrad, den Reaktionsumsatz und die Reaktorkapazität leicht ermitteln kann. Das Modell eignet sich also unmittelbar als Auslegungsbasis für Gas-Flüssigreaktoren.

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A flowingmodel for gas-liquid reactors based on the film theory

A design model for gas-liquid reactors is developed based on the film theory and under condition that the gas and liquid phase are in plug flow. An analytical solution of this system has been achieved. The mass transfer degree, the reaction conversion and the reactor capacity can be easily calculated by means of the analytical solutions. Therefore, this model can be used directly to design the gas-liquid reactors.

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Formelzeichen a _i [m ²/m ³] spezifische Phasengrenzfläche - C [kmol/m³] Konzentration - D [m²/s] Diffusionskoeffizient - F [kmol/s] Masseneinströmung der Gasphase - H [J/kmol] Henry'sche Konstante - Ha [J/kmol] Hatta-Zahl definiert in Gl. (3) - L [m] charakteristische Länge des Reaktors - Q _L [m ³/s] Volumenströmung der Flüssigphase - N [kmol/m²·s] Stoffübergangsgeschwindigkeit - p [N/m²] Partialdruck einer Komponente - p [N/m²] Gesamtdruck des Systems - r [N/m²] Strömungsstatus - x [m] Ortskoordinate längs der Diffusionsrichtung - x _A [m] Reaktionsumsatz des EduktesA - V [m³] Reaktorvolumen Griechische Buchstaben [m] Diffusionsgrenzschichtdicke - _L Flüssig-Holdup - [m] Austauschgrad - [m] Abkürzung definiert in Gl. (13) - 0 bezogen auf Anfangsstelle des Reaktors - A bezogen auf KomponenteA - b bezogen auf Bulkphase - L bezogen auf Flüssigphase - bezogen auf Einströmung - bezogen auf Ausströmung     

Injection moulding of plastics

Summary In continuation of a previous investigation a simple analytical expression is derived in closed form for the thickness distribution of the freeze-off layer which is vitrified at the (flat) wall of an oblong rectangular cavity. As has been pointed out previously, this layer is marked for amorphous polymers by the molecular orientation (birefringence pattern) in the moulded sample. One can show that a more detailed study with the aid of the coupled equations of energy and of motion will not furnish essential improvements. Problems of polymer physics like glass transition or crystallization kinetics at extreme rates of cooling and shearing must be solved first.

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Zusammenfassung In Fortsetzung einer früheren Untersuchung wurde ein einfacher analytischer Ausdruck in geschlossener Form für die Dickenverteilung der eingefrorenen Schicht abgeleitet, die an der (flachen) Wand eines langgestreckten rechteckigen Formnestes während des Einspritzvorgangs glasig erstarrt. Wie früher auseinandergesetzt wurde, wird diese Schicht bei amorphen Polymeren durch die Molekülorientierung (Doppelbrechungsmuster) im gespritzten Formteil markiert. Man kann zeigen, daß eine eingehendere Studie mit Hilfe der gekoppelten Energie- und Impulsgleichungen keine essentiellen Verbesserungen bringt. Probleme der Polymerphysik, wie Glasübergang oder Kristallisationskinetik bei extremen Abkühlungs- und Schergeschwindigkeiten, müssen erst gelöst werden.

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List of Symbols a heat diffusivity of polymer melt (averaged overT) [m²s^–1] - B breadth of mould cavity [m] - Br Brinkman number ( ) - c heat capacity of polymer melt (averaged overT) [J kg^–1 K^–1] - F ₀ Fourier number (at _i/4H ²) - h heat transfer coefficient by melt flow [J K^–1 s^–1 m^–2] - h heat transfer coefficient by layer growth [J K^–1 s^–1 m^–2] - H half height of mould cavity [m] - L length of mould cavity [m] - n exponent in eq. [18] (= 0.417) - Nu Nußelt number (2Hh/) - P pressure gradientdP/dz in mould [N m^–3] - t time [s] - t _i injection time [s] - T _g glass transition temperature of polymer [K] - T _i injection temperature of polymer melt [K] - T _l stagnation temperature [K] - T _m mould wall temperature [K] - speed of flow front during mould filling [m s^–1] - x coordinate perpendicular to mould wall [m] - z coordinate in the injection direction [m] - thickness of stagnant layer (atT _l) [m] - ₀ optically detectable freeze-off thickness [m] - ⁺ apparent layer thickness (atT _i) [m] - dimensionless freeze-off thickness (= ₀/2H) - dimensionless distance from entrance (=z/L) - _m dimensionless coordinate of layer maximum - _g dimensionless temperature (= (T _i –T _l)/(T _g –T _m)) - _i dimensionless temperature (= (T _i –T _l)/(T _i –T _m)) - _l dimensionless temperature (= (T _i –T _l)/(T _l –T _m)) - _i viscosity of polymer atT _i [N s m^–3] - _l viscosity of polymer atT _l [N s m^–3] - heat conductivity of polymer melt (averaged) [J K^–1 s^–1 m^–1] - density of polymer melt (averaged) [kg m^–3] - dimensionless time (eq. [11]) - ⁺ dimensionless parameter (eqs. [19a] and [19b]) - dimensionless layer thickness (eq. [12]) - ⁺ dimensionless parameter (eq. [20a]) - dimensionless parameter (eqs. [11a] and [11b]) Formerly at Delft University of Technology, Delft (The Netherlands).Paper presented at the Conference on Chemical Engineering Rheology, Annual Meeting of the Deutsche Rheologische Gesellschaft in Aachen, March 5–7, 1979.With 3 figures and 1 table     

General viscosity-composition relationship for dispersions,solutions and binary liquid systems

An equation for the viscosity of a mixture of two imaginary Newtonian liquids is derived. In the derivation the mathematical assumption is used that the effective activation energy for viscous flow of a binary liquid mixture is a linear combination of the reciprocals of the activation energy of the components. It contains two dependent fitting constants and has the same structure as the Mooney equation for dispersions of spherical solid particles, the Huggins equation for polymer solutions and is identical to an equation by Hoffmann and Rother, when written in the variables that the last authors used.As a consequence it can be shown that the viscosity of binary liquid mixtures, liquid resion solutions, dispersions of solid spherical particles and polymer solutions can be described very well by one and the same equation, up to the highest concentrations.It has further been found that the viscosity of dispersions of non-spherical particles, solutions of solids in organic solvents and solutions of electrolytes and non-electrolytes in water can also be described by this formula. The equation permits the construction of a straight line on which all liquids can be plotted.An algebraic analysis of the equation shows that each series of viscosity composition data can be placed in one of three rheological groups independent of the type of fraction that is used to characterize the composition.Seventy-four binary systems, covering a wide range of liquids have been used to show the applicability of the developed equation.It has been found that in most cases the data are best described by splitting them into two regions, each with its own set of dependent constants. General symbol for the fraction or concentration of the component with the higher viscosity determining the composition of a binary mixture [—] - _v Volume fraction of the component with the higher viscosity [—] - _w Weight fraction of the component with the higher viscosity [—] - _mw Molecular weight fraction of the component with the higher viscosity [—] - c Concentration of the component with the higher viscosity [g/cm³] - E ₂,E ₁,E Activation energy for viscous flow referring to the component with the higher viscosity, the lower viscosity and the viscosity of the binary mixtures, respectively [J] - ₂, ₁, Experimental parameter (with the dimension of energy) referring to the component with the higher viscosity, the component with the lower viscosity and to the binary mixtures, respectively [J] - ₁, ₂ Viscosity of the component with the lower and the higher viscosity, respectively [Pa · s] - Viscosity of a binary mixture [Pa · s] - [] The usual intrinsic viscosity of the component with the highest viscosity [cm³/g] - _r / ₁ [—] - _{sp r} – 1 [—] - [] -intrinsic viscosity [—] - [] _v Volume intrinsic viscosity [—] - [] _w Weight intrinsic viscosity [—] - [] _c Concentration intrinsic viscosity, identical to [] [cm³/g] - T _e Temperature at which the two liquids have the same viscosity [K] - _e Viscosity at temperatureT _e [Pa · s] - P ₁,P ₂ Density of the component with the lower and the higher viscosity, respectively - R Gas constant [J · Mol^–1 · K^–1]     

Bestimmung des Diffusionskoeffizienten von Wasserstoff in Wasser und wässerigen Polymerlösungen nach der CBS-Methode

Zusammenfassung Die Werte des Diffusionskoeffizienten von Wasserstoff in Wasser und wässerigen Polymerlösungen bei 20 und 30°C und ungefähr l bar Gesamtdruck werden gegeben. Die Bestimmung dieser Werte geschah nach der kürzlich veröffentlichten constant bubble-size-Methode (CBS-Methode).Der Einfluß der freien Konvektion bei der Bestimmung der Diffusionskoeffizienten von mäßig lösbaren Gasen in Flüssigkeiten ist qualitativ untersucht worden. Es wird gezeigt, daß freie Konvektion durch Erhöhung der Viskosität völlig zurückgedrängt wird. Dazu wird die Viskosität durch Zusatz eines Polymerisats erhöht.Weiterhin wurde auch der Zusammenhang zwischen Diffusionskoeffizient und zero-shear-Viskosität quantitativ untersucht. Es wurde die zero-shear-Viskosität dieser wässerigen Polymerlösungen bestimmt. Ferner ergab sich, daß der Zusammenhang zwischen dem Logarithmus des Diffusionskoeffizienten und dem Logarithmus der zero-shear-Viskosität direkt proportional war.Der Diffusionskoeffizient nimmt bei höherem Polymerzusatz leicht ab. Die experimentellen Werte wurden mit Ergebnissen aus dem Schrifttum verglichen.

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Measurement of the diffusion coefficient of hydrogen in water and aqueous polymer solutions according to the CBS-method

Values of the diffusion coefficient of hydrogen in water and aqueous polymer solutions at 20 and 30°C and about 1 bar total pressure are given. The measurement of these values has been performed according to the recently published constant bubble size method (CBS-method).The influence of free convection on the determination of diffusion coefficients of slightly soluble gases in liquids has been investigated qualitatively. It is shown that by increase of viscosity, free convection is reduced. To this end, the viscosity is increased by addition of a polymer. Furthermore, the relation between diffusion coefficient and zero-shear viscosity has been investigated quantitatively. The zero-shear viscosity of the non-Newtonian polymer solutions has been determined. A directly proportional relation between the logarithm of the diffusion coefficient and the logarithm of the zero-shear viscosity has been found.Increasing values of the polymer concentration result in a small decrease of the diffusion coefficient. The experimental values are compared with other results from literature.

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Formelzeichen a [m² s^–1] Temperaturleitfähigkeit der Flüssigkeit - A _c [m²] Oberfläche des gesperrten Kugelabschnitts - c _A [mol m^–3] Konzentration des GasesA in der Flüssigkeit - c _p [kg kg^–1] Polymerkonzentration in der Flüssigkeit - c _R [mol m^–3] Konzentration des Gases in der Flüssigkeit an der Oberfläche (r=R) - c _z [mol m^–3] Konzentration der Zusatzmenge - c [mol m^–3] Konzentration des Gases in der Flüssigkeit fürt=0 und zur fürt >0 - d [m] Gasblasendurchmesser - d _c [m] Durchmesser der Spitze des Kegelstumpfs - D _AB [m² s^–1] Diffusionskoeffizient des GasesA in der FlüssigkeitB - D _w [m² s^–1] fusionskoeffizient des Gases in der reinen Flüssigkeit - g [m s^–2] Fallbeschleunigung - L [m] Halbmesser der Innenzelle - m [s^–1] Neigung der Gerade in der Gleichung (10) - n [1] Exponent in Gleichung (12) - N _A ^* [mol] Menge des in der Flüssigkeit absorbierten GasesA - p _R [Pa] Teildruck des Gases zur=R - r [m] Kugelkoordinate - R [m] Halbmesser der Gasblase - R [Jmol^–1K^–1] Gaskonstante (R=8.314 J mol^–1 K^–1) - t [s] Zeit - T [K] Temperatur - T [K] Temperaturdifferenz - v _* ^a [m³] Volumen des in der Flüssigkeit absorbierten GasesA Griechische Formelzeichen [W K^–1 m^–2] Wärmeübergangskoeffizient - [K^–1] Wärmedehnungszahl der Flüssigkeit - [rad] Winkel - [Pa s] Viskosität - _w [Pa s] Viskosität der reinen Flüssigkeit - ₀ [Pa s] Viskosität der Polymerlösung für 0 - [Pa s] Viskosität der Polymerlösung für - [rad] Kugelkoordinate - [W K^–1 m^–1] Wärmeleitfähigkeit der Flüssigkeit - [m² s^–1] kinematische Viskosität der Flüssigkeit - _L [kg m^–3] Dichte der Flüssigkeit - [Pa m] Oberflächenspannung der Flüssigkeit - _D [m² s^–1] Standardabweichung vom Diffusionskoeffizienten - _n [1] Standardabweichung vonn - [Pa] Schubspannung Dimensionslose Kenngrößen Eö [1] Eötvössche Kenngröße (Eö=_L g R²/) - He [1] Henrysche Kenngröße (He=c _RRT/p_R) - Nu [1] Nusseltsche Kenngröße (Nu= L/) - Ra [l] Rayleighsche Kenngröße (Ra=L ³ g T/(a v))     

Die freie Oberfläche einer Flüssigkeit über einer rotierenden Scheibe

Zusammenfassung Es wird eine modifizierte Form des Weissenberg-Effekts untersucht, wobei sich die viskoelastische Flüssigkeit in einem kreiszylindrischen Gefäß befindet, an dessen Boden eine Scheibe rotiert. Normalspannungsdifferenzen rufen in der Flüssigkeit eine Strömung hervor, die auf der Drehachse von unten nach oben gerichtet ist, und die freie Oberfläche wölbt sich nahe der Achse nach außen. Unter der Voraussetzung hinreichend langsamer Strömung wird eine Theorie zweiter Ordnung entwickelt. Sie führt auf elliptische Randwertaufgaben zweiter bzw. vierter Ordnung für das Geschwindigkeitsfeld der Primärströmung in Umfangsrichtung und für die Stromfunktion der Sekundärströmung in der Meridianebene. Ihnen werden äquivalente Variationsaufgaben zugeordnet und mit der Methode der Finiten Elemente numerisch gelöst. Die Gestalt der freien Oberfläche setzt sich bei geeigneter Normierung aus drei universellen Formfunktionen zusammen, die für verschiedene Füllhöhen berechnet werden. Im experimentellen Teil wird nachgewiesen, daß durch entsprechende Messungen der Auslenkung des Flüssigkeitsspiegels die unteren Grenzwerte der beiden Normalspannungskoeffizienten bestimmt werden können. Das Rheometer besitzt den Vorzug, daß die Oberflächenspannung der Flüssigkeit die Meßgröße nur unwesentlich beeinflußt.

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Some kind of Weissenberg effect is considered where the viscoelastic fluid, being within a cylindrical vessel, is set in motion by a rotating disc near the tank bottom. Because of normal-stress differences within the fluid a secondary flow arises which is directed upwards near the axis of symmetry, and thus the free surface is deformed. Under the assumption of sufficiently slow flow a second-order theory is developed. It leads to second-order and fourth-order elliptic boundary value problems for the velocity field in azimuthal direction and for the stream function of the secondary flow, respectively. Equivalent variational problems are formulated and solved by the method of finite elements. When normalized appropriately, the shape of the free surface consists of three shape functions, which are independent of any material constants. It is shown by corresponding experiments, that the zero-shear-rate normal-stress coefficients can be determined by measuring the displacement of the free surface. In this rheometer, the surface tension of the fluid causes only insignificant influence on the quantity to be measured.

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Symbole C _H [—] Verhältnis der FormfunktionenF ₂/F₁ - f [—] die Sekundärströmung treibende radiale Volumenkraft, dimensionslos - F ₀, F₁, F₂ [—] universelle Formfunktionen - Fr [—] Froude-Zahl - g [m s^–2] Erdbeschleunigung - h [—] Auslenkung der Oberfläche, aufr ₀ bezogen - H [—] dimensionslose Füllhöhe - K [—] Kennzahl der Kapillarität - r,z [m] Zylinderkoordinaten - r, z [—] dimensionslose Koordinaten - r ₀ [m] Radius des Meßbehälters - Re [—] Reynolds-Zahl - v _r, v, v_z [m s^–1] Geschwindigkeitskomponenten - We ₁, We₂ [—] Weissenberg-Zahlen - [Pa s] Nullviskosität der Flüssigkeit - [°C] Temperatur - [m] Kapillarlänge - v ₁, v₂ [Pa s²] untere Grenzwerte der Normalspannungskoeffizienten - [kg m^–3] Dichte der Flüssigkeit - [N m^–1] Oberflächenspannung - [—] Zylinderkoordinate - [—] Dissipationsfunktion der Sekundärströmung, dimensionslos - [—] Stromfunktion, dimensionslos - [—] örtliche Winkelgeschwindigkeit, dimensionslos - [s^–1] Winkelgeschwindigkeit der Scheibe     

Heat transfer and pressure drop for purely viscous non-Newtonian fluids in turbulent flow through rectangular passages

Experimental measurements of friction factor and heat transfer for the turbulent flow of purely viscous non-Newtonian fluids in a 21 rectangular channel are compared with results previously reported for the circular tube geometry. Comparisons are also made with available analytical and empirical predictions.It is found that the rectangular duct fully established friction factor measurements are within ± 5% of the Dodge-Metzner prediction if the Kozicki generalized Reynolds number is used. A modified form of the simpler explicit equation proposed by Yoo, [i.e.f=0.079n ^0.675(Re ^*)^–0.25], is found to yield predictions for both the rectangular duct and the circular tube geometries with approximately the same accuracy as the Dodge-Metzner equation.Fully developed Stanton numbers for the rectangular duct are in good agreement with the circular tube data over a range ofn from 0.37 to 0.88 for a given Prandtl number,Pr _a , when compared at a fixed value of the Reynolds number based on the apparent viscosity evaluated at the wall shear stress. In general, the experimental data are within ± 20% of Yoo's equation,St=0.0152Re _a ^–0.155 Pr _a ^–2/3 . A new equation is proposed to bring the prediction for circular pipes as well as rectangular channels into better agreement with generally accepted Newtonian heat transfer results.

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Wärmeübergang und Druckverlust für viskose nicht-Newtonsche Fluide in turbulenter Strömung durch rechteckige Kanäle

Zusammenfassung Es werden Messungen des Reibungsfaktors und des Wärmeübergangs bei turbulenter Strömung viskoser nicht-Newtonscher Fluide in einem rechteckigen Kanal mit dem Seitenverhältnis 21 verglichen mit früheren Ergebnissen, die an runden Rohren gewonnen wurden. Weiterhin werden Vergleiche mit aus der Literatur verfügbaren analytischen und empirischen Beziehungen gemacht.Es zeigte sich, daß die Messungen des Reibungsfaktors im rechteckigen Kanal bei vollausgebildeter Strömung auf ± 5% mit der Vorhersage von Dodge-Metzner übereinstimmen, wenn die von Kozicki verallgemeinerte Reynolds-Zahl verwendet wird. Eine modifizierte Form der einfachen von Yoo vorgeschlagenen einfachen Gleichung in explizierter Form (f=0,079n ^0,675(Re ^*)^–0,25) bewies, daß sie sowohl für den rechteckigen Kanal als auch das runde Rohr die Werte mit fast der gleichen Genauigkeit wie die Methode von Dodge-Metzner vorhersagen kann.Die Stanton-Zahlen für den rechteckigen Kanal bei vollausgebildeter Strömung sind in guter Übereinstimmung mit den Werten für das runde Rohr in einem Bereich vonn= 0,37 – 0,88 für eine gegebene Prandtl-Zahl, wenn man den Vergleich bei einem vorgegebenen Wert der Reynolds-Zahl anstellt, die auf die scheinbare Viskosität — abgeleitet aus der Wandschubspannungbezogen ist. Generell läßt sich sagen, daß die Werte auf ± 20% mit der Gleichung von Yoo (St=0,0152Re _a ^–0,155 )Pr _a ^–2/3 ) übereinstimmen. Es wird eine neue Gleichung vorgeschlagen, welche sowohl die Werte für runde Rohre als auch die für rechteckige Kanäle in bessere Übereinstimmung bringt mit den in der Literatur üblichen Ergebnissen für den Wärmeübergang an Newtonsche Fluide.

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Nomenclature a constant in Eq. (8) - A area of cross-section of channel [m²] - b constant in Eq. (8) - c _p specific heat of test fluid [J kg^–1 K^–1] - d capillary tube diameter [m] - D _h hydraulic diameter, 4A/P[m] - f Fanning friction factor, _w/(g9 V²/2) - h axially local (spanwise averaged) heat transfer coefficient,q _w /(T_wi-T_b) [Wm^–2 K^–1] - k _f thermal conductivity of test fluid [Wm^–1K^–1] - K consistency index of power law fluid - n power law index - Nu fully established, local (spanwise averaged) Nusselt numberh D _h /k _f - P perimeter of channel [m] - Pr _a Prandtl number based on apparent viscosjity, c _p /k _f - Pr ^* defined as (Re _a Pr _a )/Re ^* - q _w wall heat flux [Wm^–2] - Re _a Reynolds number based on apparent viscosity, VD _h/ - Re Metzner's generalized Reynolds number in Eq. (2) - Re ^* Reynolds number defined in Eq. (8) - St Stanton number,h/( V c_p) - T _b local bulk temperature of the fluid [K] - T _wi local inside wall temperature [K] - T _wo local outside wall temperature [K] - V bulk flow velocity [m s^–1] - x distance from the inlet of channel along flow direction [m] Greek symbols shear rate [s^–1] - apparent viscosity [Pa s] - density of test fluid [kg m^–3] - shear stress [Pa] - _w shear stress at the wall [Pa] Dedicated to Prof. Dr.-Ing. U. Grigull's 75th birthday     

The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin

The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement. A cross section area of flow passage [m²] - A _mim minimum cross section area of flow passage [m²] - a width of flat tube [m] - b length of flat tube [m] - B _pT lateral pitch of flat tube: B _pT = S ₁/T _p - d _h hydraulic diameter of flow channel [m] - D _naph diffusion of naphthalene [m²/s] - f friction factor: f = pd _h/(Lu ² _max/2) - h mass transfer coefficient [m/s] - H height of winglet type vortex generators [m] - j Colburn factor [–] - JF a dimensionless ratio, defined in Eq. (23) [–] - L streamwise length of fin [m] - L _PVG longitudinal pitch of vortex generators divided by fin spacing: L _pVG = l _VG/T _p - l _VG pitch of in-line vortex generators [m] - m mass [kg] - m mass sublimation rate of naphthalene [kg/m²·s] - Nu Nusselt number: Nu = d _h/ - P pressure of naphthalene vapor [Pa] - p non-dimensional pitch of in-line vortex generators: p = l _VG/S ₂ - Pr Prandtl number [–] - Q heat transfer rate [W] - R universal gas constant [m²/s²·K] - Re Reynolds number: Re = ·u _max·d _h/ - S ₁ transversal pitch between flat tubes [m] - S ₂ longitudinal pitch between flat tubes [m] - Sc Schmidt number [–] - Sh Sherwood number [–]: Sh = hd _h/D _naph - Sr staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin) - T _p fin spacing [m] - T temperature [K] - u _max maximum velocity [m/s] - u average velocity of air [m/s] - V volume flow rate of air [m³/s] - x,y,z coordinates [m] - z sublimation depth[m] - heat transfer coefficient [W/m²·K] - heat conductivity [W/m·K] - viscosity [kg/m²·s] - density [kg/m³] - attack angle of vortex generator [°] - time interval for naphthalene sublimation [s] - fin thickness, distance between two VGs around the tube [m] - small interval - C distance between the stream direction centerlines of VGs - p pressure drop [Pa] - 0 without VG enhancement - 1, 2, I, II fin surface I, fin surface II, respectively - atm atmosphere - f fluid - fin fin - local local value - m average - naph naphthalene - n,b naphthalene at bulk flow - n,w naphthalene at wall - VG with VG enhancement - w wall or fin surface     

Crack repair using an elastic filler

The effect of repairing a crack in an elastic body using an elastic filler is examined in terms of the stress intensity levels generated at the crack tip. The effect of the filler is to change the stress field singularity from order 1/r^1/2 to 1/r^(1-λ) where r is the distance from the crack tip, and λ is the solution to a simple transcendental equation. The singularity power (1-λ) varies from (the unfilled crack limit) to 1 (the fully repaired crack), depending primarily on the scaled shear modulus ratio γ_r defined by G₂/G₁=γ_rε, where 2πε is the (small) crack angle, and the indices (1, 2) refer to base and filler material properties, respectively. The fully repaired limit is effectively reached for γ_r≈10, so that fillers with surprisingly small shear modulus ratios can be effectively used to repair cracks. This fits in with observations in the mining industry, where materials with G₂/G₁ of the order of 10^-3 have been found to be effective for stabilizing the walls of tunnels. The results are also relevant for the repair of cracks in thin elastic sheets.     

A new method for measuring time constants of a thermocouple wire in varying flow states

A new measuring method is suggested for determining the time constant of a thermocouple wire to be applied for the measurement of the true fluid temperatures in varying flow states. Based on the techniques of internal heating which are commonly used to measure mean time constants, we extend the existing method to measure instantaneous time constants continuously. A method of measurement and analysis is presented and verified experimentally.List of Symbols A _s surface area [m²] - c specific heat [J/kg K] - D diameter [m] - h heat transfer coefficient [W/m² K] - I current [A] - k thermal conductivity [W/m K] - L length [m] - r resistance per unit length [/m] - T temperature [°C] - t time [s] - t _c characteristic time to reach uniform state [s] - u velocity of stream [m/s] - V volume [m³] - x axial coordinate [m] - thermal diffusivity [m²/s] - normalized temperature (TT )/(T _RT )) - density [kg/m³] - time constant [s] - angular velocity [rad/s] - a amplitude - i initial condition - j junction of thermocouple - R reference point - surrounding The work was supported by Turbo and Power Machinery Research Center at Seoul National University and the authors are grateful to Mr. M. H. Yang for his assistance in the experiment.     

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     

Optimum rib size to enhance forced convection in a horizontal triangular duct with ribbed internal surfaces

The optimum rib size to enhance heat transfer had been proposed through an experimental investigation on the forced convection of a fully developed turbulent flow in an air-cooled horizontal equilateral triangular duct fabricated on its internal surfaces with uniformly spaced square ribs. Five different rib sizes (B) of 5 mm, 6 mm, 7 mm, 7.9 mm and 9 mm, respectively, were used in the present investigation, while the separation (S) between the center lines of two adjacent ribs was kept at a constant of 57 mm. The experimental triangular ducts were of the same axial length (L) of 1050 mm and the same hydraulic diameter (D) of 44 mm. Both the ducts and the ribs were fabricated with duralumin. For every experimental set-up, the entire inner wall of the duct was heated uniformly while the outer wall was thermally insulated. From the experimental results, a maximum average Nusselt number of the triangular duct was observed at the rib size of 7.9 mm (i.e. relative rib size ). Considering the pressure drop along the triangular duct, it was found to increase almost linearly with the rib size. Non-dimensional expressions had been developed for the determination of the average Nusselt number and the average friction factor of the equilateral triangular ducts with ribbed internal surfaces. The developed equations were valid for a wide range of Reynolds numbers of 4,000 < Re _D < 23,000 and relative rib sizes of under steady-state condition. A Inner surface area of the triangular duct [m²] - A _C Cross-sectional area of the triangular duct [m²] - B Side length of the square rib [mm] - C _P Specific heat at constant pressure [kJ·kg^–1·K^–1] - C ₁, C ₂, C ₃ Constant coefficients in Equations (10), (12) and (13), respectively - D Hydraulic diameter of the triangular duct [mm] - Electric power supplied to heat the triangular duct [W] - f Average friction factor - F View factor for thermal radiation from the duct ends to its surroundings - h Average convection heat transfer coefficient at the air/duct interface [W·m^–2 ·K^–1] - k Thermal conductivity of the air [W·m^–1 ·K^–1] - L Axial length of the triangular duct [mm] - Mass flow rate [kg·s^–1] - n ₁, n ₂, n ₃ Power indices in Equations (10), (12) and (13), respectively - Nu _D Average Nusselt number based on hydraulic diameter - P Fluid pressure [Pa] - Pr Prandtl number of the airflow - _c Steady-state forced convection from the triangular duct to the airflow [W] - _l Heat loss from external surfaces of the triangular duct assembly to the surroundings [W] - _r Radiation heat loss from both ends of the triangular duct to the surroundings [W] - Re _D Reynolds number of the airflow based on hydraulic diameter - S Uniform separation between the centre lines of two consecutive ribs [mm] - T Fluid temperature [K] - T _a Mean temperature of the airflow [K] - T _ai Inlet mean temperature of the airflow [K] - T _ao Outlet mean temperature of the airflow [K] - T _s Mean surface temperature of the triangular duct [K] - T Ambient temperature [K] - U Mean air velocity in the triangular duct [m·s^–1] - _r Mean surface-emissivity with respect to thermal radiation - Dynamic viscosity of the fluid [kg·m^–1·s^–1] - Kinematic viscosity of the airflow [m²·s^–1] - Density of the airflow [kg·m^–3] - Stefan-Boltzmann constant [W·m^–2·K^–4]