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.     

Convective heat transfer within fibrous insulation slabs

A numerical study of convective heat flow within a fibrous insulating slab is presented. The material is treated as an anisotropic porous medium and the variation of properties with temperature is taken into account. Good agreement is obtained with available experimental data for the same geometry.

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Zusammenfassung Für den konvektiven Wärmestrom in einem faserförmigen Isolierstoff wird eine numerische Berechnung angegeben. Der Stoff wird als anisotropes poröses Medium mit temperaturabhängigen Stoffwerten angesehen. Die Übereinstimmung mit verfügbaren Versuchswerten ist gut.

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Nomenclature C_p specific heat of the gas at the mean temperature - Da Darcy number=k_y/H² - Gr^* modified Grashof number=gTHk_y/²= (Grashof number) × (Darcy number) - H thickness of the specimen - P gas pressure - Pr^* modified Prandtl number= C_p/_x - Ra^* modified Rayleigh number=Gr^* Pr^* - R_p ratio of permeabilities=k_y/k_x - R_k ratio of conductivities= _y/_x - T absolute temperature of the gas - t₁ absolute temperature of the hot face - T₂ absolute temperature of the cold face - T_m mean temperature of the gas=(T₁+T₂)/2 - k_x specific permeability of the porous medium along the x-direction - k_y specific permeability of the porous medium along the y-direction - p T/T_m - q exponent - r exponent - u gas velocity along the x-direction - v gas velocity along the y-direction - X_* distance along the x-direction - y_* distance along the y-direction - T temperature difference=t₁–T₂ - thermal coefficient of expansion of the gas - _m thermal coefficient of expansion of the gas at the mean temperature - _* T–T_m - dimensionless temperature= ^*/T - _a apparent thermal conductivity of the porous medium along the x-direction - _al local apparent thermal conductivity of the porous medium along the x-direction - _x thermal conductivity of the porous medium along the x-direction in the absence of convection - _y thermal conductivity of the porous medium along the y-direction in the absence of convection - dynamic viscosity of the gas - _m dynamic viscosity of the gas at the mean temperature - kinematic viscosity of the gas - _m kinematic viscosity of the gas at the mean temperature - density of the gas - _m density of the gas at the mean temperature - ^* stream function at any point - dimensionless stream function= ^*/( _m/_m)     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress     

Finite element solutions for laminar flow and combined convection around a square prism

The finite element solutions of the full Navier-Stokes and energy equations for steady laminar flow and combined convection around square prisms with attack angles of 0° and 45° are obtained for a gas having Pr=0.7. The variations of surface shear stress, local pressure and Nusselt number are obtained over the entire prism surface including the zone beyond the point of the separation. The predicted values of drag coefficients, the location of separation, average Nusselt number and the plots of velocity flow fields and isotherms are also presented. The trend of the present numerical results seems reasonable.

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Finite-Elemente-Verfahren für laminare Strömung und kombinierte Naturkonvektion um ein quadratisches Prisma

Zusammenfassung Es wird über Lösungen der Navier-Stokesund der Energiegleichungen mit Hilfe der Finite-Elemente-Methode für stationäre laminare Strömung, kombiniert mit Naturkonvektion, um ein quadratisches Prisma berichtet, wobei als Anströmwinkel 0° und 45° gewählt wurden und Gasströmung mitPr=0,7 angenommen wurde. Die Rechnung ergibt den Verlauf der Wandschubspannungen, des örtlichen Druckes und der Nusselt-Zahl über die gesamte Oberfläche des Prismas, einschließlich des Bereiches hinter dem Ablösepunkt. Weiterhin werden in dem Aufsatz Angaben gemacht über die Widerstandskoeffizienten, die Lage des Ablösepunktes, der mittleren Nusselt-Zahl sowie der Geschwindigkeits- und Temperaturfelder. Die numerischen Ergebnisse erscheinen im Trend vernünftig zu sein.

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Nomenclature a side length of square - C _f friction drag coefficient - C _p pressure drag coefficient - C _D total drag coefficient - F _f total friction drag force - F _P total pressure drag force - Gr Groshoff number,g (T _w -T )a ³/v ² - g gravitational acceleration - h local heat transfer coefficient - K thermal conductivity - L dimensionless location of surface from the front stagna tion point,L ^*/a - L ^* dimension location of prism surface - Lc location of separation - N _j shape function - Nu, local and average Nusselt numbers - M _l shape function - P dimensionless pressure,p ^*/u ² - P ^* pressure - p _x ^* x-componentP ^* - Pe Peclet number,Re Pr - Pr Prandtl number, c/K - Ra Rayleigh number,Gr Pr - Re Reynolds number,a u /v - s direction along the sides of prism - u dimensionlessX-direction component of velocity,u ^*/u - u ^* X-direction component of velocity - u free stream velocity - dimensionlessY-direction component of velocity,v*/u - ^* Y-direction component of velocity - X X-direction axis - x dimensionlessX-direction coordinate,x ^*/a - x ^* X- direction coordinate - Y Y-direction axis - y ^* dimensionless 7-direction coordinate,y ^*/a - y ^* Y-direction coordinate Greek symbols coefficient of volumetric thermal expansion - attack angle - dynamic viscosity - kinematic viscosity,/ - density of fluid - _w dimensionless surface shear stress, _w ^* /u ² - _w ^* surface shear stress - _wx ^* x-component of _w ^/* - dimensionless temperature,      

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

On the properties of compressible gas flow in a porous media

The flow of an adiabatic gas through a porous media is treated analytically for steady one- and two-dimensional flows. The effect on a compressible Darcy flow by inertia and Forchheimer terms is studied. Finally, wave solutions are found which exhibit a cut-off frequency and a phase shift between pressure and velocity of the gas, with the velocity lagging behind the pressure.Nomenclature A area of tube for one-dimensional flow - B drag coefficient associated with Forchheimer term - c speed of sound - M Mach number - p ^* gas pressure - p dimensionless gas pressure - s coordinate along the axis of tube - t ^* time variable - t dimensionless time variable - V^* gas velocity in the porous media - V dimensionless gas velocity Greek Letters ratio of specific heat capacities - phase angle between gas pressure and velocity for linear waves - parameter indicating the importance of the inertia term - viscosity - _p natural frequency of the porous media - ^* gas density - dimensionless gas density - parameter indicating the importance of the Forchheimer term - porosity of porous media - velocity potential - stream function     

Viscous heating correction for thermally developing flows in slit die viscometry

In the thermally developing region, d _yy /dx| _y=h varies along the flow direction x, where _yy denotes the component of stress normal to the y-plane; y = ±h at the die walls. A finite element method for two-dimensional Newtonian flow in a parallel slit was used to obtain an equation relating d _yy /dx/ _y=h and the wall shear stress ₀ at the inlet; isothermal slit walls were used for the calculation and the inlet liquid temperature T₀ was assumed to be equal to the wall temperature. For a temperature-viscosity relation /₀ = [1+(T–T₀]^–1, a simple expression [(hd _yy /dx/ _y=h )/ _w0] = 1–[1-F _c(Na)] [M()+P(Pr) ·Q(Gz ^–1)] was found to hold over the practical range of parameters involved, where Na, Gz, and Pr denote the Nahme-Griffith number, Graetz number, and Prandtl number; is a dimensionless variable which depends on Na and Gz. An order-of-magnitude analysis for momentum and energy equations supports the validity of this expression. The function F _c(Na) was obtained from an analytical solution for thermally developed flow; F _c(Na) = 1 for isothermal flow. M(), P(Pr), and Q(Gz) were obtained by fitting numerical results with simple equations. The wall shear rate at the inlet can be calculated from the flow rate Q using the isothermal equation.Notation x,y Cartesian coordinates (Fig. 2) - , dimensionless spatial variables [Eq. (16)] - dimensionless variable, : = Gz(x)^–1 - dimensionless variable [Eq. (28)] - t,t ^* time, dimensionless time [Eq. (16)] - , velocity vector, dimensionless velocity vector - _x , velocity in x-direction, dimensionless velocity - _y , velocity in y-direction, dimensionless velocity - V average velocity in x-direction - _yy , ^* normal stress on y-planes, dimensionless normal stress - shear stress on y-planes acting in x-direction - _w , _w ^* value of shear stress stress at the wall, dimensionless wall shear stress - _w0, _w0 ^* wall shear stress at the inlet, dimensionless variable - , ^* rate-of-strain tensor, dimensionless tensor - wall shear rate, wall shear rate at the inlet - Q flow rate - T, T ₀, temperature, temperature at the wall and at the inlet, dimensionless temperature - h, w half the die height, width of the die - l,L the distance between the inlet and the slot region, total die length - T ₂, T ₃, T ₄ pressure transducers in the High Shear Rate Viscometer (HSRV) (Fig. 1) - P, P2, P3 pressure, liquid pressures applied to T ₂ and T ₃ - , ₀, ^* viscosity, viscosity at T = T ₀, dimensionless viscosity - viscosity-temperature coefficient [Eq. (8)] - k thermal conductivity - C _p specific heat at constant pressure - Re Reynolds number - Na Nahme-Griffith number - Gz Graetz number - Pr Prandtl number     

A noninvasive method for the measurement of flow-induced surface displacement of a compliant surface

A noninvasive optical method is described which allows the measurement of the vertical component of the instantaneous displacement of a surface at one or more points. The method has been used to study the motion of a passive compliant layer responding to the random forcing of a fully developed turbulent boundary layer. However, in principle, the measurement technique described here can be used equally well with any surface capable of scattering light and to which optical access can be gained. The technique relies on the use of electro-optic position-sensitive detectors; this type of transducer produces changes in current which are linearly proportional to the displacement of a spot of light imaged onto the active area of the detector. The system can resolve displacements as small as 2 m for a point 1.8 mm in diameter; the final output signal of the system is found to be linear for displacements up to 200 m, and the overall frequency response is from DC to greater than 1 kHz. As an example of the use of the system, results detailing measurements obtained at both one and two points simultaneously are presented.List of symbols C _t elastic transverse wave speed = (G/)^1/2 - d ⁺ spot diameter normalized by viscous length scale - G frequency average of G() - G() shear storage modulus - G() shear loss modulus - l. viscous length scale = v/u _* - N total number of sampled data values - r separation vector for 2-point measurements = (, ) - rms root-mean-square value - R momentum thickness Reynolds number = U _t8/v - t time - u (y) mean streamwise component of velocity in boundary layer - u _* friction velocity = (t _w/)^1/2 - U free-stream velocity - x, y, z longitudinal, normal and spanwise directions - y _o undisturbed surface position - vertical component of compliant surface displacement - ₉₉ boundary layer thickness for which u(y) = 0.99 U _t8 - _l viscous sublayer thickness 5 l _* - frequency average of G()/ - boundary layer momentum thicknes = - fluid dynamic viscosity - v fluid kinematic viscosity = / - , longitudinal, spanwise components of separation vector r - fluid density - time delay - _w wall shear stress     

A mixing length model for turbulent flow in constant cross section ducts

In this paper, a study is made of the damping influence of the wall on turbulent fluid flow. By considering the oscillation of the whole of the boundary, van Driest's original hypothesis has been extended to obtain the wall damping factor in flow in a duct of constant cross section. The damping factor is used in conjunction with mixing length expressions to obtain the velocity field. Particular examples considered are plane parallel flow and axisymmetric flow in a pipe and in an annulus.

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Ein Modell für die Mischungslänge von turbulenten Strömungen in Rohren mit konstantem Querschnitt

Zusammenfassung In dieser Arbeit wurde der dämpfende Wandeinfluß in turbulenten Strömungen untersucht. Unter Berücksichtigung der Schwingungen in der gesamten Grenzschicht wurde die ursprüngliche Theorie von van Driest erweitert und ein Dämpfungsfaktor an der Wand in Rohrströmungen mit konstantem Querschnitt ermittelt. Dieser Dämpfungsfaktor diente in Verbindung mit Ausdrücken für die Mischungslänge zur Bestimmung des Geschwindigkeitsfeldes. Ausgewählte Beispiele waren die ebene Parallelströmung sowie die Zylinderströmung in einem Rohr und einem Ringspalt.

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Nomenclature A, A^*(=Au/v) Parameter defined in text - b, b^*(=bu/v) semi-width of parallel plate channel - c(= 1/A) parameter defined in text - E[, /2] complete elliptic integral of the second kind - d damping factor - F, G, H functions - l, l^*(=/v) mixing length - M_O, _O functions - r, r^*(=ru/v) radius - A real part of function - R, S, T, U functions - u, u^*(=u/u) velocity in flow direction Z - friction velocity - x, y, z co-ordinates (z in flow direction) - y^*(=yu/v) non-dimensional wall distance - fluid density - , _eff kinematic viscosity, effective kinematic viscosity - phase angle, or polar coordinate angle - shear stress - (=r/r_W) radius ratio - angular velocity Suffixes w wall value - far from a wall     

Einige Beziehungen zwischen Transportkoeffizienten und thermodynamischen Zustandsgrö\en

Zusammenfassung Eine früher mitgeteilte Beziehung [1] zwischen der ViskositÄt und dem isenthalpen Joule-Thomson-Koeffizienten _h wird für kleine Drücke theoretisch begründet und an sieben Stoffen nachgeprüft. Die WärmeleitfÄhigkeit wird als Funktion von c_v für drei Stoffe dargestellt.

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Some relations between transport coefficients and thermodynamical properties

A formerly given relation [1] between viscosity and isenthalpic Joule-Thomson-coefficient _h is proved theoretically for small pressures and checked with seven substances. The heat conductivity is presented as a function ofc_v for three substances.

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Bezeichnungen B ^* dimensionsloser zweiter Virialkoeffizient - B ₁ ^* Ableitung vonB ^* nachT ^*.B ₁ ^*=T ^* dB ^*/dT ^* - c _v isochore spezifische WärmekapazitÄt - C _p ^o isobare molare WärmekapazitÄt fürp 0 - h spezifische Enthalpie - k Boltzmann-Konstante.k=R/N _A - M molare Masse - N _A Avogadro-Konstante - p Druck - R molare Gaskonstante - R _i spezifische Gaskonstante des Stoffesi - it Celsius-Temperatur - T Kelvin-Temperatur - T ^* dimensionslose Temperatur.T ^*=kT/ - _h isenthalper Joule-Thomson-Koeffizient. _h=(T/p)_h - , Konstanten der Potentialfunktion - ViskositÄt - WärmeleitfÄhigkeit - (^2,2)^* dimensionsloses Sto\integral     

Viskoelastisches Verhalten von ABS-Polymeren in der Schmelze

Zusammenfassung Mit Hilfe eines Schwingungsviskosimeters mit konzentrischen Zylindern wurde der komplexe SchubmodulG +iG von ABS-Polymeren bei Frequenzen zwischen 10^–3 und 50 Hz und Temperaturen zwischen 130 und 250 °C gemessen. Bei hohen Frequenzen ergeben sich keine wesentlichen Unterschiede im Verlauf der Modulkurven, verglichen mit homogenen Schmelzen. Das viskoelastische Verhalten wird hier vor allem durch das Verschlaufungsnetzwerk der kohärenten Phase bestimmt. Bei tiefen Frequenzen verhalten sich ABS-Polymere in der Schmelze dagegen ähnlich wie vernetzte Kautschuke:G wird frequenzunabhängig, steigt proportional zu ·T an und nimmt wesentlich größere Werte an alsG. Es überwiegen also die elastischen Eigenschaften, während die Schmelzen homogener Polymerer bei tiefen Frequenzen vorwiegend viskos sind. Dieses gummielastische Verhalten ist um so ausgeprägter, je höher der Kautschukgehalt, der Pfropfungsgrad der Kautschukteilchen und, bei gleichem Kautschukgehalt, die Teilchenzahl ist.AusG und G läßt sich die komplexe Schwingungsviskosität ^* berechnen, deren Betrag ¦^*¦ bei vielen Kunststoffschmelzen mit der Viskositätsfunktion () bei stationären Scherströmungen übereinstimmt. Bei ABS-Polymeren wird ¦^*¦ bei tiefen Frequenzen nicht konstant, sondern steigt mit abnehmender Frequenz stark an. Es existiert also offensichtlich keine konstante Nullviskosität ₀ wie bei homogenen Schmelzen.Ein ähnliches viskoelastisches Verhalten wie ABS-Polymere, wenn auch schwächer ausgeprägt, zeigen Kunststoffe mit anorganischen Füllstoffen wie TiO₂.

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Summary The complex shear moduliG +iG of ABS-polymers were measured by means of a dynamic viscometer with concentric cylinders at frequencies between 10^–3 and 50 cps and temperatures between 130 and 250 °C. At high frequencies there are no remarkable differences in the shape of the modulus curves compared with homogeneous melts. The viscoelastic behaviour is here mainly determined by the entanglement network of the coherent phase.At low frequencies molten ABS-Polymers behave like crosslinked rubbers:G becomes independent of frequency, is proportional to ·T and has much greater values thanG. That means that the elastic properties are prevailing, whereas the melts of homogeneous polymers are mainly viscous at low frequencies. This rubberlike behaviour is the more marked, the higher the rubber contents, the degree of grafting of the rubber particles and, with equal rubber contents, the number of particles.FromG andG the complex dynamic viscosity ^* can be evaluated. For many polymer melts the absolute value ¦^*¦ corresponds to the steady-state viscosity (). For ABS-polymers ¦^*¦ does not become constant at low frequencies but rises to much higher values with decreasing frequency. Obviously there is no constant zero — shear viscosity as there is for homogeneous melts.A similar viscoelastic behaviour as shown by ABS-polymers, though less marked, is shown by plastics with anorganic fillers like TiO₂.

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Den Herren Dr.Haaf, Dr.Heinz und Dr.Stein danke ich für die Herstellung der Proben.     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) 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 - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - 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, m - t time, s - 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 - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

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     

Dynamic melt viscoelastic behavior of random copolymers

Summary The viscoelastic properties of 65/35 styrenen-butyl methacrylate random copolymers were determined using the Eccentric Rotating Disks device of the Rheometrics Mechanical Spectrometer. Similar to the behavior observed in homopolymers, an increase in the molecular weight of the copolymer resulted in extension of the rubbery plateau and in a reduction in the terminal region. The dynamic complex viscosity showed onset of non-Newtonian flow at higher frequencies, with the non-Newtonian region increasing with increasing molecular weight.The elastic modulus,G, was dependent upon the frequency,, asG ^1.5 in the terminal region, rather than asG ² observed for polystyrene. The viscous modulus,G, was proportional to the frequency,, asG , similar to what is observed for polystyrene. The dynamic viscosity | ^*| at high frequencies showed a region independent of molecular weight where a power law of | ^*| ^0.9 is applicable, consistent with entanglement models. Thy dynamic viscosity at low frequencies in the Newtonian region is related to molecular weight as |^*| . Using WLF equations, the coefficient of expansion, _f , was obtained that, together with glass transition, showed a negative deviation from the Fox-Flory relationship.

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Zusammenfassung Die viskoelastischen Eigenschaften von statistischen 65/35-Styrol/n-Butyl-Methacrylat-Kopolymeren wurden mit Hilfe einer Maxwell-Rheometer-Anordnung in Verbindung mit dem Mechanischen Spektrometer der Fa. Rheometrics bestimmt. Ähnlich dem bei Homopolymeren beobachteten Verhalten ergab sich auch hier mit wachsendem Molekulargewicht eine Verbreiterung des Kautschuk-Plateaus und eine Verkleinerung des Endbereichs. Die komplexe Viskosität zeigte erst bei höheren Frequenzen das Einsetzen nicht-newtonschen Fließens an, wobei der nichtnewtonsche Bereich mit steigendem Molekulargewicht größer wurde.Der SpeichermodulG ergab sich im Endbereich als proportional zu ^1,5, im Unterschied zu der bei Polystyrol beobachteten Proportionalität mit ². Dagegen war der VerlustmodulG der Frequenz direkt proportional, ähnlich wie es auch bei Polystyrol beobachtet worden war. Die dynamische Viskosität | ^*| zeigte unabhängig vom Molekulargewicht bei hohen Frequenzen einen Bereich, in dem eine Potenz-Beziehung | ^*| ~ ^0,9 herrschte, was auf die Wirkung von Verzweigungen hindeutet. Dagegen galt bei den niedrigen Frequenzen des newtonschen Bereichs|^*| ~ . Mit Hilfe der WLF-Gleichung wurde der Ausdehnungskoeffizient _f bestimmt, der ebenso wie der Glasübergang eine negative Abweichung von der Fox-Flory-Beziehung zeigte.

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With 10 figures and 1 table     

A method for the analysis of recirculating turbulent flows in a cavity

The complex fluid-dynamic aspects of a turbulent recirculating flow in a cavity with axial throughflow, and a rotating wall, were investigated by adopting a simple procedure for evaluating the turbulent stresses. The flow field was divided into two regions, a core and a wall region respectively. A wall function was adopted in the zones near to the solid boundaries, while a constant eddy diffusivity was assumed, in the core, following the indications of computed heat transfer coefficients in comparison with existing experimental data. The distributions of the stream function and of the tangential velocity are presented for a range of the rotational Reynolds number of the rotating wall and of the Reynolds number of the throughflow.

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Turbulente Rezirkulationsströmung in einem Hohlraum

Zusammenfassung Die komplizierten fluiddynamischen Aspekte einer turbulenten Rezirkulationsströmung in einem Hohlraum mit axialem Durchfluß und einer rotierenden Wand werden unter Verwendung einer vereinfachten Methode zur Berechnung der turbulenten Spannungen betrachtet. Das Strömungsfeld wird in einen Kern und einen Wandbereich aufgeteilt. Für die wandnahen Zonen wird eine Wandfunktion angenommen, während im Kern mit konstanter Wirbeldiffusivität gerechnet wird, was durch den Vergleich berechneter mit gemessenen Wärmeübergangskoeffizienten gerechtfertigt erscheint. Verteilungen der Stromfunktion und der tangentialen Geschwindigkeit sind für einen bestimmten Bereich der Reynoldszahlen für die Wandrotation und der für den Durchfluß angegeben.

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Nomenclature L axial length of enclosure - P dimensionless pressure, p^*2 - p static pressure - R dimensionless radial coordinate, r/r^* - r radial coordinate - r^* reference length, equal to r_O for enclosure - r_i radii of inlet and exit apertures - Re Reynolds number, v^*r^*/ - Re_i pipe Reynolds number, ¯v_zi(2r_i)/ - Re_t turbulent Reynolds number, Re(/) - Re rotational Reynolds number, r ₀ ² / - t dimensionless time,t/(r^*/v^*) - t time - V_r, V, Vz dimensionless velocity components, V_r/v^*, v, v_z/v^* - v_i turbulent fluctuation of the i-component of velocity - v_r, v, v_z velocity components - v^* reference velocity, equal to ¯v_zi for enclosure - X coordinate along a wall, x/r^* - Y coordinate normal to a wall, y/r^* - Z dimensionless axial coordinate, z/r^* - z axial coordinate - eddy diffusivity for momentum - dynamic viscosity - kinematic viscosity - density - shear stress - dimensionless shear stress, /v^*2 - dimensionless stream function, /r*²v*² - stream function - angular velocity - tangential vorticity component - ()_eff effective - ()_l laminar - ()_t turbulent - mean over the time     

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     

Über Differentialgleichungen für Wärme- und Stoffaustausch von hygroskopischen Stoffen

Zusammenfassung Die beiden Differentialgleichungssysteme vonKrischer undLykow werden miteinander verglichen. Dabei ergibt sich, daß die in der deutschen und russischen Literatur angewandten mathematischen Modelle der Trocknung von kapillarporösen Körpern praktisch übereinstimmen. Es werden die Transformationsgleichungen der dimensionslosen Kenngrößen angegeben, die die Beziehungen zwischen den beiden Systemen herstellen.

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The differential equations ofKrischer andLuikow for unsteady internal heat and mass transfer in the porous medium are compared. It is shown, that the mathematical models for drying in the German and Russian literature are equivalent. The transform relations of the non-dimensional parameters between the two models are given.

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Formelzeichen nach Krischer z laufende Koordinate in Strömungsrichtung in m - R kennzeichnende Abmessung des Körpers in m - t Zeit in h - Raumgewicht bei mittlerer Feuchtigkeit in kg/m³ - _w Teilgewicht des Wassers in 1 m³ Trockengut in kg/m³ - _wa Anfangsfeuchtigkeit in kg/m³ - _D Dampfdichte in kg/m³ - _L Luftvolumen je m³ Trocknungsgut in m³/m³ - Temperatur in °C - _u Umgebungstemperatur in °C - _a Anfangstemperatur in °C - r Verdampfungswärme in kcal/kg - q _E Wärmeentwicklung in kcal/m³ h - c spezifische Wärmekapazität des Trockengutes in kcal/kg grd - Wärmeleitfähigkeit in kcal/m h grd - Feuchtigkeitsleitzahl des Trockengutes in m²/h - wirksame Diffusionszahl von Wasserdampf in Luft in m²/h - Diffusionswiderstandszahl des Trockengutes — - Konstante — - Konstante in kg/m³ grd Formelzeichen nach Lykow X=r/R dimensionslose Koordinate des Körpers;r laufende Koordinate in m;R kennzeichnende Abmessung in m; - Fo=a/R ² Fourier-Zahl;a Temperaturleitzahl in m²/h; Zeit in h - T(X, Fo)=t(r, )– 0/t dimensionslose Temperatur des Körpers im Punkt mit KoordinateX für den ZeitpunktFo;t(r, ) Temperatur in °C; 0 mittlere Anfangstemperatur in °C; t ein vorher angenommener Temperaturunterschied in grd - (X, Fo)= ₀–u(r, )/u dimensionsloses Potential des Stoffübergangs im Punkt mit KoordinateX für den ZeitpunktFo;u(r, ) Feuchtigkeitsgehalt des Trockengutes in kg/kg; ₀ mittlerer Anfangsfeuchtigkeitsgehalt in kg/kg; u ein vorher angenommener Unterschied des Feuchtigkeitsgehalts in kg/kg - Ko= u/c t Kosowitsch-Zahl; Verdampfungswärme in kcal/kg;c spezifische Wärmekapazität des Trockengutes in kcal/kg - Ko*=Ko modifizierte Kosowitsch Zhal; Kenngröße der Zustandsänderung - Pn= t/u Posnowsche Zahl;=a _T ^m /a _m Thermogradientkoeffizient in 1/grd;a _T ^m thermische Stoffübergangszahl (charakterisiert den Stoffstrom unter der Einwirkung von Temperaturgradienten) in m²/h grd;a _m Stoffübergangszahl (charakterisiert den Stoffstrom unter der Einwirkung von Feuchtigkeitsgradienten) in m²/h - Lu=a _m/a Lykowsche Zahl - Ki _q=q _q ()·R/ _q t dimensionsloser Wärmestrom (Kirpitschew-Zahl);q _q() Wärmestrom durch die Körperoberfläche in kcal/m²; _q Wärmeleitfähigkeit in kcal/m² h grd - Ki _m=q _m ()·R/a _{m 0} u dimensionsloser Stoffstrom;q _m() Stoffstrom durch die Körperoberfläche in kg/m² h; ₀ Wichte des Trockengutes in kg/m³     

Laminar stability analysis of condensate film flow

The linear stability theory is used to study stability characteristics of laminar gravity-induced condensate film flow down an arbitrarily inclined wall. The coupled equations describing the velocity and temperature disturbances are solved numerically. The results show that laminar condensate films are unstable in all practical situations. Several stabilizing effects are acting on the film flow; these are: the angle of inclination, the surface tension at large wave numbers, the condensation rate at small Reynolds numbers, and to a certain extent the Prandtl number. For a vertical plate, the expected wavelengths of the disturbances are presented as functions of the Reynolds numbers of the condensate flow.

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Zusammenfassung Mit Hilfe der linearen StabilitÄtstheorie werden die StabilitÄtseigenschaften laminarer Kondensatfilme an ebenen WÄnden untersucht. Die Gleichungssysteme, die Temperatur- und Geschwindigkeitsstörungen beschreiben, werden numerisch gelöst. Es zeigt sich, da\ Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einflu\ von OberflÄchenspannung, Schwerkraft und Stoffübertragung durch Kondensation werden diskutiert. Für eine senkrechte Wand werden die zu erwartenden WellenlÄngen der Störungen als Funktion der Reynoldszahlen des Kondensatfilms angegeben.

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Abrreviations

Nomenclature C^*=C _r ^* + iC _i ^* dimensional complex wave velocity - C=C^*/u₀ dimensionless wave velocity - c_p specific heat at constant pressure - g gravitational acceleration - h_n defined by Eq. (16) - h_fg latent heat - k thermal conductivity - Pe=PrRe Peclet number - Pr Prandtl number - P_y defined by Eq. (15) - q iaPe - Re=u₀ Reynolds number - S temperature disturbance amplitude - t^* dimensional time - t=t^* u₀/ dimensionless time - T dimensional temperature - T_s saturation temperature - T_w wall temperature - T =T_s–T_w temperature drop across liquid film - u^*, v^* dimensional velocity component - v=v^*/u₀ dimensionless velocity components - u₀ dimensional surface velocity of undisturbed film flow - x^*, y^* dimensional coordinates - x=x^*/ dimensionless coordmates - Y_n functional vector defined by Eq. (20) Greek Symbols dimensionless wave number - roots of Eq. (20) - n defined by Eq. (21) - local thickness of undisturbed condensate film - ^* wavelength, dimensional - wavelength, dimensionless - temperature variable - kinematic viscosity of liquid - liquid density - _g vapor density - surface tension - stream function disturbance amplitude - stream function - angle of inclination     

Das erweiterte Korrespondenzgesetz für die dynamische Idealviskosität und die Idealwärmeleitfähigkeit reiner Stoffe

Zusammenfassung Zur Berechnung der dynamischen Idealviskosität Ideal (T) und der Idealwärmeleitfähigkeit ideal (T) benötigt man die kritische TemperaturT _kr, das kritische spezifische Volum _kr, die MolmasseM, den kritischen Parameter _kr und die molare isochore WärmekapazitätC _v(T). Sowohl das theoretisch, als auch das empirisch abgeleitete erweiterte Korrespondenzgesetz ergeben eine für praktische Zwecke ausreichende Genauigkeit für die Meßwertwiedergabe, die bei den assoziierenden Stoffen und den Quantenstoffen jedoch geringer ist als bei den Normalstoffen.

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The extended correspondence law for the ideal dynamic viscosity and the ideal thermal conductivity of pure substances

For the calculation of the ideal dynamic viscosity Ideal (T) and the ideal thermal conductivity ideal (T) the critical temperatureT _kr, the critical specific volumev _kr, the molecular massM, the critical parameter _kr, and the molar isochoric heat capacityC _v(T) is needed. Not only the theoretically determined but also the empirically determined extended correspondence law gives for practical use a good representation of the measured data, which for the associating substances and the quantum substances is not so good as for the normal substances.