Mass transport and reaction in catalyst pellets

When heterogeneous chemical reactions take place in porous catalysts, mass transport can occur by bulk diffusion, Knudsen diffusion, and convective transport. Previous studies of these phenomena have been largely based on Maxwell's dusty gas model with the convective transport or Darcy flow added to the diffusive transport. This is done in order to satisfy one of the limiting conditions encountered in the study of flow in porous media. A more fundamental approach consists of the use of the method of volume averaging and the general form of the species momentum equation. For an N-component system, this leads to N independent flux relations to be used in conjunction with the volume-averaged species continuity equations.Roman Letters _A (t) surface area of a species body, m² - a _v interfacial area per unit volume, m^-1 - A _e area of entrances and exits for the -phase contained within the averaging volume, m² - A _K area of the - surface contained within the averaging volume, m² - b _A species A body force, N/kg - b mass average body force, N/kg - B inverse tortuosity tensor for bulk diffusion - c total molar concentration, moles/m³ - c _A species A molar concentration, moles/m³ - _A surface concentration of species A, moles/m² - C_A² intrinsic phase average molar concentration, moles/m³ - c _A – C_A², spatial deviation concentration, moles/m³ - c _A mean molecular speed for species A, m/s - binary diffusion coefficient, m²/s - D _A ^{K, eff} Knudsen diffusion coefficient for species A, m²/s - f vector that maps P _A into P _A , m - g gravitational vector, m/s² - G second order tensor that maps N _A into N _A for free molecule flow conditions - H inverse tortuosity tensor for Knudsen diffusion - I unit tensor - j _A c _A u _A ^* , molar diffusive flux, moles/m²s - K Darcy's Law permeability tensor, m² - L macroscopic length scale, m - L _D diffusive length, m - l characteristic length for the -phase, m - l _A mean free path for species A, m - M _A molecular weight of species A, kg/mole - n outwardly directed unit normal vector - n _K unit normal vector directed from the -phase toward the -phase - n outwardly directed unit normal vector at the entrances and exits of the -phase contained within the averaging volume - N _A c _A v _A molar flux of species A, moles/m²s - N _A intrinsic phase average of the species A molar flux, moles/m²s - \~N _A spatial deviation of the molar flux of species A, moles/m²s - p total pressure, N/m² - P p + , total pressure over and above the hydrostatic pressure, N/m² - P _A partial pressure of species A, N/m² - p _A intrinsic phase average partial pressure, N/m² - P_A – p _A, spatial deviation partial pressure, N/m² - P _A p_A + _AA partial pressure of species A over and above the hydrostatic pressure of species A, N/m² - p _ab diffusive force exerted by species B on species A, N/m³ - universal gas constant, N m/moles K - R _A molar rate of production of species A owing to homogeneous chemical reaction, moles/m³s - molar rate of production of species A owing to heterogeneous chemical reaction, moles/m²s - r _A mass rate of production of species A owing to homogeneous chemical reaction, kg/m³s - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - t _A species stress vector, N/m² - T _A species stress tensor, N/m² - T total stress tensor, N/m² - T temperature, K - T spatial average temperature, K - u _A v _A – v, mass diffusion velocity, m/su _A ^* v_A – v^*, molar diffusion velocity, m/s - u _o velocity of the rigid, solid phase relative to some inertial frame, m/s - v _A species velocity, m/s - v mass average velocity, m/s - v ^* molar average velocity, m/s - v _A ^* species velocity of those molecules of species A generated by chemical reaction, m/s - _A (t) volume of a species A body, m³ - averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ - v phase average, mass average velocity, m/s - w arbitrary velocity vector, m/s - x _A c _A /c mole fraction of species A - X _A intrinsic phase average mole fraction - X _A – X _A , spatial deviation mole fraction Greek Letters V/V volume fraction of the -phase - _A sum of all terms in the species A momentum equation that are small compared to the diffusive force, N/m³ - viscosity of the -phase, Ns/m² - _A mass density of species A, kg/m³ - total mass density, kg/m³ - _a species viscous stress tensor, N/m² - total viscous stress tensor, N/m² - tortuosity factor - total body force potential function, Nm/kg - _a species body force potential function, Nm/kg - 3.1416 - _{a a} / mass fraction of species A     

Experimental and numerical results on transient heat transfer to supercritical helium at low temperatures

Transient heat transfer coefficients to a forced flow supercritical helium at low temperatures have been measured and compared with data of a numerical computer simulation. The helium flow through the cooling tubes was described in the simulation by a two dimensional model. The helium properties were stored as a function of enthalpy and pressure in look up tables.The experimental and numerical results agree well. At this moment the numerical code is a good instrument for computing the thermal hydraulic behaviour of hollow superconductors, cooled by a flow of supercritical helium, to get an impression on stability and cooling performance.

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Experimentelle und numerische Lösungen für transienten Wärmetransport von überkritischem Helium bei niedrigen Temperaturen

Zusammenfassung Es wurden transiente Wärmeübertragungskoeffizienten einer erzwungenen Strömung von überkritischem Helium bei niedrigen Temperaturen gemessen und verglichen mit Daten einer numerischen Computersimulation. Der Heliumstrom durch die Kühlrohre wurde in der Simulation von einem zweidimensionalen Modell beschrieben. Die Eigenschaften des Heliums wurden als eine Funktion von Enthalpie und Druck gespeichert. Die experimentellen und numerischen Ergebnisse stimmen gut überein. Folglich ist das numerische Verfahren ein gutes Instrument das thermisch-hydraulische Verhalten von hohlen Supra-Leitern, gekühlt von einem Strom überkritischen Heliums, zu berechnen, um einen Eindruck von Stabilitäts- und Kühlleistungen zu bekommen.

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Nomenclature A m² surface - a m²/s thermal diffusivity - c _p J/kg K specific heat - D m (hydraulic) diameter of the test tube - H J/kg enthalpy — in flow - J/m³ enthalpy — in tube - h W/m² K heat transfer coefficient - L _m m mixing length - m kg mass of the test tube - P N/m² pressure - R m radius of the tube - r m radial coordinate in flow - RRR residual resistance ratio(_e, 300 K/_e, 4,2 K) - S W/m³ source term of heat - T K temperature - t s time - U m/s axial velocity - V m/s radial velocity - x m axial coordinate in tube - y m R–r, the distance from the wall - y⁺ - Z m axial coordinate in flow - N s/m² viscosity - _T N s/m₂ turbulent viscosity - J/m K thermal conductivity - kg/m³ density - _e m specific electrical resistivity - _w N/m² wall shear stress - W heat flow     

Transport coefficients of air in the region of temperatures from 3000 to 25,000 ° K and pressures of 0.1, 1, 10, and 100 atm

In the article a calculation is made of the transport properties of air in the region of temperatures from 3000 to 25,000 °K and pressures of 0.1, 1, 10, and 100 atm. The calculations are made within the framework of molecular-kinetic theory in a first approximation of the distribution function by the Chapman-Enskog method, taking account of the first four terms of a series expansion by Sonin polynomials. The collision integrals _ij ^l,s of the air components (N₂-N₂, O₂-O₂, NO-NO, N₂-O₂, N₂-NO, O₂-NO, N₂-N, N₂-O, O₂-N, O₂-O, N-NO, O-NO, N-N, O-O, O-N, e-N, e-O, e-N₂, e-O₂, e-NO, N-N⁺, O-O⁺, N-O⁺, O-N⁺, electron-ion, ion-ion, electron-electron) of the ordersl=1, 4 and s=l, (8-l) are calculated. As far as possible the collision integrals presented are calculated on the basis of experimentally measured interaction potentials obtained through the direct scattering of atomic and molecular beams on gas targets or other beams. The missing information on interaction potentials is borrowed from works on spectroscopic analysis. The scattering of an electron on an oxygen atom is calculated in the framework of a quantum-mechanical analysis. The collision integrals presented for charged components of the air were calculated numerically for the Debye-Hückel screening potential. The collision integral moments obtained in the work can be used to calculate the kinetic coefficients for an arbitrary mixture of nitrogen and oxygen, including the case of a variable elementary composition.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 80–90, March–April, 1973.In conclusion the author thanks V. G. Sevast'yanenko under whose guidance the work was carried out.     

Dichotomy Spectrum for Nonautonomous Differential Equations

For nonautonomous linear differential equations x=A(t) x with locally integrable A: RR ^N×N the so-called dichotomy spectrum is investigated in this paper. As the closely related dichotomy spectrum for skew product flows with compact base (Sacker–Sell spectrum) our dichotomy spectrum for nonautonomous differential equations consists of at most N closed intervals, which in contrast to the Sacker–Sell spectrum may be unbounded. In the constant coefficients case these intervals reduce to the real parts of the eigenvalues of A. In any case the spectral intervals are associated with spectral manifolds comprising solutions with a common exponential growth rate. The main result of this paper is a spectral theorem which describes all possible forms of the dichotomy spectrum.     

Rheology of polydisperse polymers: relationship between intermolecular interactions and molecular weight distribution

An expression of the relaxation function of linear polydisperse polymers is proposed in terms of intermolecular couplings of reptative chains. The relaxation times of each molecular weight are assumed to be shifted according to a tube renewal mechanism accounting for the diffusion of the surrounding chains. The subsequent shift is applied to the relaxation function of each molecular weight obtained from an analytical expression of the complex compliance J ^*(). Therefore the complex shear modulus G ^*() is derived from the overall relaxation function using the probability density accounting for the molecular weight distribution and four species-dependent parameters: a front factor A for zero-shear viscosity, plateau modulus G _N ⁰ , activation energy E and characteristic temperature T . All the main features of the theology of polydisperse polymers are described by the proposed model.     

Velocity and turbulence measurements in combustion systems

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

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Investigation of active media for a short-wave gas dynamic laser

The process of formation of an active medium on the visible and ultraviolet intervals in a recombining plasma flow through a supersonic nozzle has been investigated theoretically and experimentally. The conditions of onset of inversion over its interval of existence have been theoretically determined. A highly ionized xenon plasma, generated by a pulsed gas dynamic source of the shock tube with nozzle type, was used in the experiments. Amplification of the emission in the blue—green region of the spectrum was obtained for the 6p ⁴ D ₅ ^2/° —6s ⁴ P _3/2 (wavelength 0.5419 m) and 6p' ² P _3/25d ² D _5/2(0.4973m) transitions of the Xe II ion.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 165–173, March–April, 1992.     

Pore structures and transport properties of sandstone

To quantitatively analyze the macroscopic properties of the flow in porous media by means of the continuum approach, detailed information (velocity and pressure fields) on the microscopic scale is necessary. In this paper, the numerical solution for incompressible, Newtonian flow in a diverging-converging representative unit cell (RUC) is presented. A new solution procedure for the problem is introduced. A review of the accuracy of the computational method is given.Nomenclature A _ff ^* area of entrance and exit of RUC - A _fs ^* interfacial area between the fluid and solid phases - d throat diameter of RUC (m) - D pore diameter of RUC (m) - i, j unit vector for RUC - L ^* wave length of a unit cell - L _p pore length of RUC (m) - L _t throat length of RUC (m) - n unit outwardly directed vector for the fluid phase - p ^* fluid pressure - ^* cross-sectional mean pressure - _en ^* entrance cross-sectional mean pressure - Re _d Reynolds number - x ^*, r^* cylindrical coordinates - u ^*, v^* velocity - u _cl ^* centerline velocity - _d mean velocity at the throat of RUC (m/s) - _D mean velocity at the large segment of RUC (m/s) Greek viscosity coefficient (Ns/m²) - _p excess momentum loss factor defined in (4.1) - fluid density (kg/m³) - ^* stream function - ^* vorticity - dimensionless circulation defined in (2.7) Symbols - the mean value - * dimensionless quantities     

Wärmeübergang bei freier Konvektion unter dem Einfluß von elektrolytisch erzeugten Wasserstoffblasen

Zusammenfassung Die Arbeit befaßt sich mit dem Wärmeübergang von einer waagrechten Heizfläche an wäßrige NaOH-Lösungen, um zu klären, ob der Wärmeübergang bei einphasiger freier Konvektion und beim Blasensieden durch an der Heizfläche elektrolytisch erzeugte Gasblasen verbessert werden kann. Hierzu wurden Messungen zwischen 60°C und Siedetemperatur bei Umgebungsdruck ausgeführt, wobei sich die Wärmestromdichte von 4,78·10⁴ W/m² bis 2,10·10⁵ W/m² und die elektrische Stromdichte von 0 bis 2100 A/m² erstreckten. Um Stoffwerte des Wassers durch Zugabe von NaOH nicht wesentlich zu beeinflußen, wurde die Lösungskonzentration bis höchstens 0,25 mol/l entsprechend 10 g/l variiert. Die Messungen ergaben eine Verbesserung des Wärmeübergangs durch elektrolytisch erzeugte Gasblasen im Vergleich zu dem ohne elektrolytische Blasenbildung bis zum Faktor 6. Die Verbesserung nimmt mit steigender Lösungstemperatur und steigender elektrischer Stromdichte zu. Eine höhere Wärmestromdichte führt zwar zu einer Zunahme des Wärmeübergangskoeffizienten . Gleichzeitig nimmt jedoch das Verhältnis /₀ ab, wenn ₀ der Wärmeübergangskoeffizient ohne elektrolytische Blasenbildung ist. Der Einfluß der Lösungskonzentration auf den Wärmeübergang ist im untersuchten Konzentrationsbereich vernachlässigbar klein.

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Heat transfer in free convection under the influence of electrolytically generated hydrogen bubbles

The paper deals with heat transfer from a horizontal heating surface to weak aqueous solutions of NaOH in order to explain whether the heat transfer in natural convection and pool boiling can be enhanced by hydrogen bubbles generated electrolytically at the heating surface. Measurements were made at liquid temperatures between 60°C and saturation temperature at atmospheric pressure. The heat flux density ranged from 4.78·10⁴ W/m² to 2.10·10⁵ W/m² and the current density from 0 to 2100 A/m². In order not to essentially change the physical properties of water by addition of NaOH, the concentration of the solution was varied only up to 0.25 mol/l. The experiments showed an enhancement of heat transfer up to a factor of 6 due to the electrolytically produced hydrogen bubbles. The enhancement of heat transfer increases with increasing solution temperature and with increasing current density. An increasing heat flux density leads to an increase of the heat transfer coefficient . At the same time the ratio /₀ decreases, where ₀ is the heat transfer coefficient without hydrogen evolution. The effect of concentration on heat transfer coefficients can be neglected in the concentration range covered by the experiments.

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Formelzeichen a Temperaturleitfähigkeit [m²/s] - C Konstante - E ₀ reversible Zersetzungsspannung von Wasser [V] - i elektrische Stromdichte [A/m²] - g Fallbeschleunigung [m/s²] - Gr Grashof-Zahl (Gr= g l ³/ ²) - K ₁ Kennzahl (K ₁=i E ₀/q) - K ₂ Kennzahl (K ₂=p _s/p) - l charakteristische Länge [m] - m, n Exponenten - Nu Nußelt-Zahl (Nu= l/) - p Systemdruck [MPa] - p _s Dampfdruck [MPa] - Pr Prandtl-Zahl (Pr=/a) - q Wärmestromdichte [W/m²] - Wärmeübergangskoeffizient [W/m² K] - ₀ Wärmeübergangskoeffizient ohne Elektrolyse [W/m² K] - räumlicher Wärmeausdehnungskoeffizient [1/K] - Wärmeleitfähigkeit [W/K m] - Temperatur [°C] - treibende Temperaturdifferenz [K] - kinematische Viskosität [m²/s] Herrn Prof. Dr.-Ing., U. Grigull zum 80. Geburtstag gewidmet     

The interaction of thermal conduction and infrared gaseous radiation

Summary The present investigation is concerned with energy transfer in gases which transmit heat by the combined mechanisms of molecular conduction and infrared radiation. For illustrative purposes an analysis is presented for a gas bounded by two parallel black plates and within which there is a uniform heat source. Radiation-conduction interaction parameters, appropriate to both the optically thin and large path length limits, are presented for a number of gases. Numerical solutions for the gas centerline temperature have been obtained for CO, CO₂, H₂O, and CH₄, and these are compared with limiting solutions.Nomenclature A _i total band absorptance of the ith band, cm^–1 - A _oi correlation quantity, cm^–1 - _i dimensionless band absorptance, A _i /A _oi - c speed of light - C _oi ^/2 correlation quantity, atm^–1 cm^–1 - C _oi ² Planck's function evaluated at center of ith band - h Planck's constant - k Boltzmann's constant - L distance between plates, cm - P gas pressure, atm - q _R radiation heat flux, W/cm² - Q heat source or sink, W/cm³ - T temperature, °K - T ₁ wall temperature - T _e centerline temperature - u _i dimensionless coordinate, C _oi ^/2 Py - u _oi dimensionless path length, C _oi ^/2 PL - y physical coordinate, cm - dimensionless coordinate, y/L - thermal conductivity, W (cm °K)^–1     

Measurement of the turbulent diffusivity in the near field of variable density jets using conditional velocimetry

Pulsed laser Mie scattering and laser Doppler velocimetry (LDV), both conditioned on the origin of the seed particles, have been successively performed in turbulent jets with variable density. In the early stages of the jet developments, significant differences are measured between the ensemble average LDV data obtained by jet seeding and those obtained by seeding the ambient air. Careful analysis of the marker statistics shows that this difference is a quantitative measure of the turbulent mixing. The good agreement with gradient–diffusion modelling suggests the validity of a general diffusion equation where the velocities involved are expressed in terms of ensemble conditional Favre averages. This operator accounts for all events (including intermittent ones) and for variations in the density of the marked fluid whose velocity is still specified by the binary origin of the marker.List of symbols D_L laminar diffusivity, m²/s - D_T turbulent diffusivity, m²/s - d diameter of the jet nozzle, m - F_r Froude number - J diffusion vector, m/s - k global sensitivity of the detection system for one particle (signal level) - N_P number of seed particles in the probe volume - N_P,i number of seed particles in sample i - N_P⁽ⁱ⁾ value of N_P in channel i - N_B number of Doppler bursts - count rate of bursts, s^–1 - N_v number of validated Doppler bursts - count rate of validated bursts, s^–1 - N_id number of ideal particles - N_id* number of marked ideal particles - P* probability that an ideal particle be marked by a seed particle - P(z) probability density function for z, m³/kg - probability to have k seed particles in the probe volume - probability of having k seed particle conditioned on a given value of z - r radial coordinate, m - R =⁽¹⁾/⁽²⁾, density ratio - S₁ local signal level with jet seeding - S₁⁽¹⁾ reference signal level in pure stream 1 with jet seeding - s₁ = S_1/S₁⁽¹⁾, normalized signal - v_c volumic capacity of the probe volume, m³ - V velocity vector, m/s - V_x axial velocity component, m/s - V_r radial velocity component, m/s - V_P particulate velocity vector, m/s - V_Pj velocity vector of particle j, m/s - V_Pij velocity vector of the jth particle in sample i, m/s - V_i velocity vector of the marked flow for realization i, m/s - V_1,i velocity vector of the flow such it is marked in realization i by particles issuing only from stream 1, m/s - x axial coordinate, m - Y_i local mass fraction of species i - Z mixture fraction:local mass fraction of jet fluid - Z_i mixture fraction for realization iGreek local density, kg/m³ - _i local density for realization i, kg/m³ - ⁽¹⁾ density in stream 1 (density of the jet fluid), kg/m³ - ₁ time of flight of jet seed particles to reach the probe volume, s - _B duration of a Doppler burst, sAverages <A> ensemble average of A - Ā time average of A - Favre average, , ( ) the present notation is only due to printing problems - A Favre fluctuation,      

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     

The third-normal stress difference in entangled melts: Quantitative stress-optical measurements in oscillatory shear

Stress-optical measurements are used to quantitatively determine the third-normal stress difference (N ₃ = N ₁ + N ₂) in three entangled polymer melts during small amplitude (<15%) oscillatory shear over a wide dynamic range. The results are presented in terms of the three material functions that describe N ₃ in oscillatory shear: the real and imaginary parts of its complex amplitude ₃ ^* = ₃ - i ₃ , and its displacement ₃ ^d . The results confirm that these functions are related to the dynamic modulus by ² ₃ ^* ()=(1-)[G ^*())– G ^*(2)] and ² ₃ ^d ()=(1- )G() as predicted by many constitutive equations, where = –N ₂/N ₁. The value of (1-) is found to be 0.69±0.07 for poly(ethylene-propylene) and 0.76±0.07 for polyisoprene. This corresponds to –N ₂/N ₁ = 0.31 and 0.24±0.07, close to the prediction of the reptation model when the independent alignment approximation is used, i.e., –N ₂/N ₁ = 2/7 – 0.28.     

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

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

Vakuum-Wärmedämmung bei Stahlmantel-Fernheizrohren

Zusammenfassung Zur Zeit gewinnen neue Entwicklungen von Fernheizrohr-Verlegungssystemen an Bedeutung, mit denen die Investitions- und Betriebskosten herabgesetzt und die Lebensdauer und Betriebssicherheit heraufgesetzt werden können. In diesem Zusammenhang steht diese Untersuchung zur Wärmedämmung eines Vakuum-Stahlmantelrohres, bei dem sich durch Druckabsenkung im Ringraum die effektive Wärmeleitfähigkeit der faser- oder pulverartigen Isoliermaterialien herabsetzen läßt (Smoluchowski-Effekt). Die Ergebnisse zur effektiven Wärmeleitfähigkeit der Isolierungen zeigen, daß die Werte bei 1 mbar etwa 30 bis 60% und bei 0,1 mbar noch etwa 15 bis 25% der Werte bei Atmosphärendruck betragen.

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Heat insulation in vacuum steel-jacket pipe systems

New developments in district heating supply, which lower the investment and operation costs and increase the service life and operational safety are gaining greater significance. In this connection stands this investigation of the heat insulation in vacuum steel-jacket pipe systems, in which the pressure reduction in the closed ring cavity lower the effective conductivity of the fibrous or porous insulating materials (Smoluchowski effect). The results for the values of the effective thermal conductivity of the insulations are at 1 mbar only 30 to 60% and at 0.1 mbar approximately 15 to 25% of the values at atmospheric pressure.

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Formelzeichen a 1 Anteil der hintereinander geschalteten Feststoffelemente - A m² Fläche - A l/s Konstante (Gl. (28)) - B ₀ m² Permeabilitätskoeffizient - c J/(kg K) spez. Wärmekapazität - c inp J/(kg K) spez. Wärmekapazität bei konstantem Druck - c _v J/(kg K) spez. Wärmekapazität bei konstantem Volumen - d m Moleküldurchmesser - d m Probeplattendicke in der Plattenapparatur - d _p m Partikeldurchmesser - g m/s² Erdbeschleunigung - K J/K Boltzmann-Konstante - K W/(m·K²) Konstante - l m Länge - M kg/mol Molmasse - p N/m²; mbar Druck - Q ^* W Wärmestrom - Q _H ^* W Heizleistung in der Rohrapparatur - Q ^p/* W Heizleistung in der Plattenapparatur - r m Radius - R J/(kmol·K) universelle Gaskonstante - s m wirksamer Faser- bzw. Partikelabstand - t s Zeit - T K, °C Temperatur - T _heiß K, °C Temperatur der heißen Oberfläche - T _kalt K, °C Temperatur der kalten Oberfläche - T _m K, °C Mitteltemperatur=1/2 (T ₁+T ₃) bzw. 1/2(T _p+T _k) - T _p °C Heizplattentemperatur - T _K °C Kühlplattentemperatur - ^* V m³/s Volumenstrom - 1/K Temperaturausdehnungskoeffizient - 1 Akkomodationskoeffizient - 1 Emissionsverhältnis des Isoliermaterialfeststoffes - kg/(m·s) dynamische Viskosität - x 1 Isentropenexponent (x=c _p /c _v ) - _eff W/(mK) effektive Wärmeleitfähigkeit eines Isoliermaterials - ₁ W/(mK) ₁ gemessene effektive Wärmeleitfähigkeit eines Isoliermaterials - _k W/(mK) äquivalente Wärmeleitfähigkeit infolge freier Konvektion - _Lp W/(mK) druckabhängige Wärmeleitfähigkeit eines Gases (Luft) zwischen engen Begrenzungswänden - _Lo W/(mK) Wärmeleitfähigkeit eines Gases (Luft) im freien Gasraum - _R W/(mK) äquivalente Wärmeleitfähigkeit infolge Strahlung - _s W/(mK) Wärmeleitfähigkeit des Feststoffmaterials - X _S W/(mK) Wärmeleitfähigkeit des Feststoffgerippes - _w W/(mK) äquivalente Wärmeleitfähigkeit infolge Gas-Feststoff-Wechselwirkungen - m mittlere freie Weglänge eines Gasmoleküls - l/ ₁ m²K/W Wärmedurchgangswiderstand eines Isoliermaterials - 1 Porosität - kg/m³ Dichte - _s, W/(m² K⁴) Strahlungskonstante des schwarzen Körpers - Nu _k 1 Nusselt-Zahl für Konvektion - Ra ₀ 1 Rayleigh-Zahl (Gl. (15)) - Gr ₀ 1 Grashof-Zahl (Gl. (15)) - Pr 1 Prandtl-Zahl (Gl. (15)) Die hier vorgestellte Forschungsarbeit wurde mit Mitteln des BMFT und der Firmen Dillinger Stahlbau GmbH, Fernwärme Niederrhein GmbH, Kabelmetall electro GmbH, Preussag AG und Winterrohrbau finanziert.     

Lower critical pressure for dynamic stability of a spherical shell

Dynamic stability of a thin spherical shell is investigated analytically under a uniform normal pressure.The purpose of this paper is to present a dynamic stability criterion which together with the energy method result and the numerical integration of the asymptotic nonlinear shell equations permit to find a closed form analytic expression for the lower critical pressure.The dynamic stability criterion states that the change in kinetic energy is equal to the work of all the forces between the initial and the buckled position after the dynamic stage of buckling.The solution of this nonlinear problem can be interpreted as the trajectory of a material point moving in a nonconservative force field.The resulting lower critical pressure curve lies along all the lowest known experimental data. It determines the boundary for the absolute dynamic stability and can be very useful for the practical shell design to prevent buckling.Nomenclature A constant value 2.2 - dA _s elemental area of the surface of the shell - a, b constants of integration - C elasticity modulus of elastic foundation - C ₁, C ₂, C ₃ constants of integration - D Eh ³/12(1– ²), stiffness of the shell - e base of natural logarithms - E modulus of elasticity - F force vector - F _x , F _y components of the force vector in the x, y directions - h thickness of the shell - H height of a segment of a shell - In *³ A - i imaginary number - I moment of inertia of a beam - K _s , K changes of curvature in the s, and directions - M bending moment in a beam - M _s ,M shell moment resultants - N _s ,N ,N _s shell membrane resultants - N _s ⁰ initial value of the membrane force in the meridional direction - P external uniform normal pressure - p _cl classical value of critical pressure (obtained by Von Kármán from the linear analysis of the shell) - Q shell transverse shear resultant - R initial radius of curvature of the middle surface - r distance from the point on the shell to the axis of symmetry of the shell - S ₀ ⁴ =DR ²/Eh characteristic length of the shell - s distance measured along the meridian of the shell - t= – ^* distance measured from the transition zone - new variables to study the behavior of the shell in the vicinity of x=1 - V velocity vector - Vol total change of volume of the shell - relative change of the curvature in the direction - non-dimensional membrane force - Z=u _x +iu _y complex variable - w deflection of the beam - dW _b bending energy per unit surface of the shell - W _b change of bending energy of the shell - W _c energy of initial uniform compression of the shell - dW _m membrane energy per unit surface of the shell - W _m change of membrane energy - W _p total work done by the external pressure - W _T total work of the buckled shell - _s , membrane deformations - initial angle of the shell - angle of the deformed shell - Poisson's ratio Part of this research was carried out at Princeton University.     

Structure and rheology of styrene methyl methacrylate copolymers

Summary In a series of homopolymers and copolymers of styrene (S) and methyl methacrylate (MMA) (100/0, 85/15, 50/50, 15/85, 0/100),T _g ,G and | ^*| increased significantly when MMA was increased beyond 50%. Although the magnitude ofJ _e ⁰ asymptote was the same, it occurred at decreasing frequencies with increasing MMA. TheJ increases linearly withT —T _g, but more so with polystyrene (PS) than with polymethyl methacrylate (PMMA). The | ^*| showed non-Newtonian behavior at decreasing frequency with increasing MMA, the asymptotic form being consistent withGraessley models. The viscoelastic behavior can be correlated with chain flexibility and SMS triad concentration in copolymers.

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Zusammenfassung Man findet, daß bei einer Reihe von Homopolymerisaten und Copolymerisaten von Styrol (S) und Methylmethacrylat (MMA) (100/0, 85/15, 50/50, 15/85, 0/100)T _g, G und | ^*| erheblich anwachsen, wenn der MMA-Gehalt 50% übersteigt. Obgleich die Größe des asymptotischen Wertes vonJ _e ⁰ konstant bleibt, wird dieser mit wachsendem MMA-Gehalt schon bei kleineren Frequenzen annähernd erreicht.J wächst linear mitT —T _g, jedoch bei Polystyrol (PS) stärker als bei Polymethylmethacrylat (PMMA). Der Betrag der komplexen Viskosität | ^*| zeigt mit wachsendem MMA-Gehalt von einer kleiner werdenden Frequenz ab ein nicht-newtonsches Verhalten an, wobei der asymptotische Verlauf einer Modellvoraussage nachGraessley folgt. Das viskoelastische Verhalten kann mit der Kettenflexibilität und der Konzentration der SMS-Triaden in den Kopolymerisaten korreliert werden.

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With 6 figures and 4 tables     

The microscopic analysis of high forchheimer number flow in porous media

High Forchheimer number flow through a rigid porous medium is numerically analysed by means of the volumetric averaging concept. The microscopic flow mechanisms, which must be known in order to understand the macroscopic flow phenomena, are studied by utilising a periodic diverging-converging representative unit cell (RUC). The detailed information for the microscopic flow field, in association with the locally averaged momentum balance, makes it possible to quantitatively demonstrate that the microscopic inertial phenomenon, which leads to distorted velocity and pressure fields, is the fundamental reason for the onset of nonlinear (non-Darcy) effects as velocity increases. The hydrodynamic definitions for Darcy's law permeabilityk, the inertial coefficient and Forchheimer number Fo are obtained by applying the averaging theorem to the pore level Navier-Stokes equations. Finally, these macroscopic parameters are numerically calculated at various combinations of micro-geometry and flow rate, and graphically correlated with the relevant microscopic parameters.Nomenclature a _i body force acceleration (m/s²) - A viscous integral term defined in (4.6) - A _f area of entrance and exist of RUC (m²) - A _fs interfacial area between the fluid and solid phases (m²) - B pressure integral term defined in (4.4) - d throat diameter of RUC (m) - D pore diameter of RUC (m) - Fo Forchheimer number defined in (4.1) and (4.10) - g gravitational acceleration (m/s²) - i, j microscopic unit vector for RUC - k Darcy's law permeability (m²) - k _v velocity dependent permeability defined in (4.1) (m²) - L length of a unit cell (m) - L _p pore length of RUC (m) - L _t throat length of RUC (m) - n unit outwardly directed vector for the fluid phase - p microscopic fluid pressure (N/m²) - P macroscopic fluid pressure (N/m²) - _en mean pressure at entrance of RUC (N/m²) - _ex mean pressure at exit of RUC (N/m²) - r _i,r coordinate on the macroscopic scale (m) - Re _d Reynolds number defined in (4.5) - u _i,u microscopic velocity (m/s) - specific discharge (m/s) - _d mean velocity at the throat of RUC (m/s) - v microscopic velocity (m/s) - V _b representative elementary volume (REV) (m³) - V _f volume occupied by the fluid within REV (m³) - V _s volume occupied by the solid within REV (m³) - x _i,x coordinate on the microscopic scale (m) - X _i,X coordinate on the macroscopic scale (m) Greek the inertia coefficient (1/m) - viscosity coefficient (Ns/m²) - _i microscopic unit vector - areosity at the entrance and the exit cross-section of RUC - fluid density (kg/m³) - porosity - _f a general property of the fluid phase Symbols ^f intrinsic phase average - the fluctuating part of _f - the mean value of _f - _f ^* the dimensionless value of _f     

Influence of cavitation on blade characteristics

An experimental study of flow around a blade with a modified NACA 4418 profile was conducted in a water tunnel that also enables control of the cavitation conditions within it. Pressure, lift force, drag force and pitching moment acting on the blade were measured for different blade angles and cavitation numbers, respectively. Relationships between these parameters were elaborated and some of them are presented here in dimensionless form. The analysis of results confirmed that cavitation changes the pressure distribution significantly. As a consequence, lift force and pitching moment are reduced, and the drag force is increased. When the cavitation cloud covers one side of the blade and the flow becomes more and more vaporous, the drag force also begins to decrease. The cavity length is increased by increasing the blade angle and by decreasing thé cavitation number.List of symbols A (m²) blade area,B ·L - B (m) blade width - C _D (–) drag coefficient,F _D /(p _d ·A) - C _L (–) lift coefficient,F _L /(P _d ·A) - C _M (–) pitching moment coefficient,M/(P _d ·A ·L) - C _p (–) pressure coefficient, (p-p _r )/p _d - F (N) force - L (m) blade length - M (Nm) pitching moment - p (Pa) local pressure on blade surface - p _d (Pa) dynamic pressure, ·V ²/2 - p _r (Pa) reference wall pressure at blade nose position if there would be no blade in the tunnel - p _v (Pa) vapor pressure - p ₁ (Pa) wall pressure 350 mm in front of thé blade axis - Re (–) Reynolds number,V ·L/v - V (m/s) mean velocity of flow in the tunnel - x (m) Cartesian coordinate along thé blade profile cord - x _c (m) cavity length,x-coordinate of cavity end - (°) blade angle - v (m²/s²) kinematic viscosity - (kg/m³) fluid density - (–) cavitation number, (p _r –p _v )/p _d - (°) angle of tangent to thé blade profile contour