Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

Two-phase brine mixtures in the geothermal context and the polymer flood model

Two-phase mixtures of hot brine and steam are important in geothermal reservoirs under exploitation. In a simple model, the flows are described by a parabolic equation for the pressure with a derivative coupling to a pair of wave equations for saturation and salt concentration. We show that the wave speed matrix for the hyperbolic part of the coupled system is formally identical to the corresponding matrix in the polymer flood model for oil recovery. For the class ofstrongly diffusive hot brine models, the identification is more than formal, so that the wave phenomena predicted for the polymer flood model will also be observed in geothermal reservoirs.Roman Symbols A,B coefficient matrices (5) - c(x,t) salt concentration (primary dependent variable) - C(p, s, c, q _t) wave speed matrix (6) - f source term (5) - g acceleration due to gravity (constant) - h _b(p, c) brine specific enthalpy - h _v(p) vapour specific enthalpy - j conservation flux (1) - k absolute permeability (constant) - k _b(s), k_v(s) relative permeabilities of the brine and vapour phases - K conductivity - p(x,t) pressure (primary dependent variable) - q volume flux (Darcy velocity) (3) - s(x,t) brine saturation (primary dependent variable) - t time (primary independent variable) - T=T _sat(p) saturation temperature - u _b(p, c) brine specific internal energy - u _m T rock matrix specific internal energy - u _v(p) vapour specific internal energy - U(x, t) shock velocity - x space (primary independent variable) Greek Symbols porosity (constant) - _b(p, c) brine dynamic viscosity - _v(p) vapour dynamic viscosity - (p, s, c) conservation density (1) - _b(p, c) brine density - _v(p) vapour density Suffixes b brine - m rock matrix - t total - v vapour - S salt - M mass - E energy     

An elementary proof of the polar factorization of vector-valued functions

We present an elementary proof of an important result of Y. Brenier [Br1, Br2], namely, that vector fields in ^d satisfying a nondegeneracy condition admit the polar factorization (*) u(x)=(s(x)), where is a convex function and s is a measure-preserving mapping. Brenier solves a minimization problem using Monge-Kantorovich theory; whereas we turn our attention to a dual problem, whose Euler-Lagrange equation turns out to be (*).     

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x=f _o(X)+f ₁(X,t), where f _o(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.     

Saint-Venant's principle in linear two-dimensional elasticity for non-striplike domains

Let be a simply connected domain in the x ₁-x ₂ plane which lies within the strip 0<x ₂, is a simple closed piecewise smooth curve. Let l= [(x ₁, x ₂): (x ₁, x ₂) and x ₁>0], _l = [(x ₁ x ₂): (x ₁ ,x ₂) and x ₁>1>0].Suppose that a two-dimensional homogeneous isotropic elastic body occupies , that a self-equilibrated stress loading is applied to - l, and that l is stress-free. Knowles [2] and Flavin [6] showed that the elastic energy in _l decays exponentially with respect to l with an exponential decay constant of the form k/b, where k is a universal constant. It is shown here that a decay constant of the form c/ may be obtained where c is a universal constant and is a characteristic dimension of , which is more appropriate than b for general non-striplike domains. In addition, an appropriate decay theorem is obtained for coil-like domains.     

Construction of a General Solution of a Degenerate Linear System of Differential Equations with Irregular Singular Point

We investigate the asymptotic behavior of the general solution of the linear system of differential equations with irregular singular point

Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem

Conclusions We have investigated solutions of equation (3) when ² is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for 0 and ² = _i ² we have exactly two nontrivial solutions which bifurcate to the right of _i ² ; these solutions are shown to exist in an interval ( _i ² , _i ² + ₀). The method of Section 3 may then be used to extend these two solutions to the right of _i ² + ₀ providing that ²= _i ² + ₀ is not an eigenvalue of the linear operator (13) evaluated at = ± ₁. Either a solution can be uniquely extended, or there exists a value of ²where the bifurcation method must be applied again³.While the method used here gives the exact number of solutions bifurcating from _i ² , other problems remain open; for example, it is still not proven that the two bifurcating branches have i zeros, as is the case for Hammerstein operators with oscillation kernels [4]. The conjecture of Odeh and Tadjbakhsh that there are exactly 2(i+1) nontrivial solutions in the interval _i ² < _i ^+1/2 remains un-answered, although it would be proven if one could show that there is no secondary bifurcation as in the cases of Kolodner [7] and Coffman [8].     

Experimental investigation of the asymptotically approached Mach reflection over the second surface in a double wedge reflection

The assumption that the Mach reflection which is formed over the second surface of a double wedge with angles _w ¹ and kw/2 approaches asymptotically the Mach reflection which would have been obtained by an identical incident shock wave over a single wedge with an angle _w = _w ² was verified experimentally. The verification of this assumption supports the shock polar analysis suggested by Ben-Dor et al. (1987) for the study of the reflection process of a planar shock wave over a double wedge. Measurements of the rate of approach to the asymptotic value are also provided.     

Streamwise pseudo-vortical structures and associated vorticity in the near-wall region of a wall-bounded turbulent shear flow

Streamwise pseudo-vortical motions near the wall in a fully-developed two-dimensional turbulent channel flow are clearly visualized in the plane perpendicular to the flow direction by a sophisticated hydrogen-bubble technique. This technique utilizes partially insulated fine wires, which generate hydrogen-bubble clusters at several distances from the wall. These flow visualizations also supply quantitative data on two instantaneous velocity components, and w, as well as the streamwise vorticity, _x . The vorticity field thus obtained shows quasi-periodicity in the spanwise direction and also a double-layer structure near the wall, both of which are qualitatively in good agreement with a pseudo-vortical motion model of the viscous wall-region.List of symbols C _i ,c _i ,d _i constants in Eqs. (2), (3) and (4) - H channel width (m) - Re _H Reynolds number (= U _c H/) - Re Reynolds number (= U _c /) - T period (s) - t time (s) - U mean streamwise velocity (m/s) - U _c center-line velocity (m/s) - u friction velocity (m/s) - u, , w velocity fluctuations (m/s) - x, y, z coordinates (m) - ^* displacement thickness (m) - momentum thickness (m) - mean low-speed streak spacing (m) - kinematic viscosity (m²/s) - phase difference - _x streamwise vorticity fluctuation (1/s) - ( )⁺ normalized by u and - (^–) root mean square value - (^–) statistical average This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Heat transfer from laminar flow in ducts with elliptic cross-section

Summary The cooling of a hot fluid in laminar Newtonian flow through cooled elliptic tubes has been calculated theoretically. Numerical data have been computed for the two values 1.25 and 4 of the axial ratio of the elliptic cross-section . For =1.25 the influence of non-zero thermal resistance between outmost fluid layer and isothermal surroundings has also been investigated. Special attention has been given to the distribution of heat flux around the perimeter; when increases the flux varies more with the position at the circumference. This positional dependence becomes less pronounced, however, as the (position-independent) thermal resistance of the wall increases.Flattening of the conduit, while maintaining its cross-sectional area constant, improves the cooling. Comparison with rectangular pipes shows that this improvement is not as marked with elliptic as with rectangular pipes.Nomenclature A _k =A _{m, n} coefficients of expansion (6) - a, b half-axes of ellipse, b<a - a _p =a _{r, s} coefficients of representation (V) - D hydraulic diameter, = 4S/P; S = cross-sectional area, P = perimeter - D _e equivalent diameter, according to (13) - n coordinate (outward) normal to the tube wall - T temperature of fluid - T _i temperature of fluid at the inlet - T _s temperature of surroundings - v ₀ mean velocity of fluid - v _z longitudinal velocity of fluid - x, y carthesian coordinates coinciding with axes of ellipse - z coordinate in flow direction - , dimensionless half-axes of ellipse, =a/D and =b/D - _t heat transfer coefficient from fluid at bulk temperature to surroundings; equation (11) - _w heat transfer coefficient at the wall; equation (3) - axial ratio of ellipse, = a/b = / - , , , dimensionless coordinates; =x/D, =y/D, =z/D, =n/D - dimensionless temperature, = (T–T _s)/(T _i–T _s) - ₀ cup-mixing mean value of ; equation (10) - thermal conductivity of fluid - _m,n = _k eigenvalue - c volumetric heat capacity of fluid - _{m, n} = _k = _k eigenfunction; equations (6) and (I) - Nu total Nusselt number, = _t D/ - Nusselt number at large distance from the inlet - Nu _w wall Nusselt number, = _w D/, based on _w - Pé Péclet number, = ₀ Dc/     

On hydromagnetic fluctuating flow past an infinite porous plate in slip flow regime

The effect of a uniform external magnetic field on the laminar, incompressible rarefied gas flow along an infinite porous flat plate is studied under the following conditions: 1) there is uniform suction, 2) the external flow velocity varies periodically with time in magnitude but not in direction, 3) the magnetic Reynolds number is small and 4) the current occurs under slip flow boundary conditions. Expressions for the velocity and temperature fields in the boundary layer are obtained. The response of skin friction, and heat transfer to the fluctuating stream is studied for variations in the rarefaction parameter h ₁, the magnetic field parameter M, and the frequency of the fluctuating stream.Nomenclature c _p specific heat of the gas - f ₁ Maxwells reflection coefficient - f ₂ thermal accommodation coefficient - G as defined in (36) - h ₁ rarefaction parameter (L ₁ v ₀/) - h ₂ nondimensional temperature jump coefficient (L ₂ v ₀/) - H amplitude of the skin friction - k thermal conductivity - K _n Knudsen number - L mean free path - L ₁ (2–f ₁/f ₁) L - L ₂ - M magnetic field parameter ( ₀ B ₀ ² /v ₀ ² ) - m 1/2[1+(1+4M+4i)^1/2], m _r+im _i - n ₁ 1/2[1+(1+4M)^1/2] - q heat flux - R suction Reynolds number - T temperature - x, y coordinates along and perpendicular to the plates - u, v velocity components along x, y-directions - density - kinematic viscosity - ₀ electrical conductivity - Prandtl number - frequency of the fluctuating stream - nondimensional frequency parameter (/v ₀ ² ) - nondimensional distance from wall (v ₀ y/) - phase lead - U _{0 0} mean velocity in the boundary layer - U _{0 1}, U _{0 2} amplitude of the velocity fluctuation in the boundary layer - specific heat ratio     

Finite element simulation of solidification problems

The modelling of liquid-solid phase change phenomena is extremely important in many areas of science and engineering. In particular, the solidification of molten metals during various casting methods in the foundry, provides a source of important practical problems which may be resolved economically with the aid of computational models of the heat transfer processes involved. Experimental design analysis is often prohibitively expensive, and the geometries and complex boundary conditions encountered preclude any analytical solutions to the problems posed. Thus the motivation for numerical simulation and computer aided design (CAD) systems is clear, and several mathematical/computational modelling techniques have been brought to bear in this area during recent years.This paper reports on the application of the finite element method to solidification problems, principally concerning industrial casting processes. Although convective heat transfer has been modelled, the work herein considers only heat conduction, for clarity. The heat transfer model has also been coupled with thermal stress analysis packages to predict mechanical behaviour including cracking and eventual failure, but this is reported elsewhere.Following the introduction, the mathematical and computational modelling tools are described in detail, for completeness. A discussion on the handling of the phase change interface and latent heat effects is then presented. Some aspects of the solution procedures are examined next, together with special techniques for dealing with the mold-metal interface. Finally, some numerical examples are presented which substantiate the capabilities of the finite element model, in both two and three dimensions.Nomenclature c heat capacity - C capacitance matrix - f time function - F loading term - h heat convection coefficient - H specific enthalpy - |J| Jacobian determinant - || patch approximation to |J| - k thermal conductivity - K conductance matrix - L latent heat - unit outward normal - N _i nodal shape function - q known heat flux - R _i nodal heat capacity - S phase change interface - t time - T temperature - known boundary temperature - T vector of nodal temperatures - T _a ambient temperature - T _c solidification temperature - T _L liquidus temperature - T ₀ initial temperature - T _s solidus temperature - x space coordinates - interface heat transfer coefficient - iteration parameter - boundary of domain - T solidification range - t timestep magnitude - vector gradient operator - convergence tolerance - timestepping parameter - t known vector in alternating-direction formulation - Laplace modifying parameter - (, ) local space coordinates - density - time limit - () shape function factor - () shape function factor - domain of interest     

Space correlation measurements in a forced shear layer with wall injection

Comprehension of wall-injection flow in a channel in the presence of different geometric discontinuities is necessary as part of the general investigations concerning combustion instabilities of solid propellant rocket motors. In order to characterise the aerodynamic flow field and to evaluate the influence of an obstacle inside a porous channel in such a case, experimental studies were conducted on a 1/40 scale model of the new ARIANE V motor. In fact, the flow is only induced by wall-injection and the presence of an obstacle creates a particular shear layer development in the obstacle wake. Particular attention was given to the unstable dynamic conditions of the shear layer. A thermal seeding of the shear layer was made in order to qualify the heat transfer therein, and especially to emphasise the turbulent structure development. Transverse and longitudinal spatial correlations were measured to characterise turbulence scales in the shear layer. At the origin of the shear layer, the decay of turbulence memory is found to be similar to that observed in a forced flow boundary layer, but the injecting wall modifies the change in structures. The wall flow is found to preserve the turbulent structures in such a way that the turbulence memory predominates in a longitudinal direction.List of Symbols f frequency (Hz) - h channel height (m) - h _v obstacle height (m) - L mean dimension of large structures (m) - P pressure (Pa) - m mass flow rate (kg/s) - r distance between probes in X direction (m) - r ^+(–) distance between probes in Y direction (m) - u longitudinal velocity in X direction (m/s) - v transversal velocity in Y direction (m/s) - T temperature (K) - x,y,z axis system (m) - t = T-T _amb (K) - v kinematic viscosity (mVs) - density (kg/m³) - characteristic porous size (urn) - _u longitudinal rms (m/s) - I _u dynamic turbulence intensity _u /u _max - I _T thermal turbulence intensity - M Mach number u/a - Re _w wall Reynolds number v _w h/v - Re _c longitudinal Reynolds number u _c h/v - R _uT thermal dynamic correlation coefficient - St Strouhal number fh/u - X,Y,Z axis system normalised by the channel height h - X _S longitudinal position of the obstacle - X X-X _S - amb ambient reference - c flowing cavity - fav head end - l lateral direction - g longitudinal direction max maximum at a longitudinal position - w wall The authors thank the CNES for its financial support, and in particular E. Robert and R. Bee.     

Plane entry flows and energy estimates for the navier-stokes equations

One of the classic problems of laminar flow theory is the development of velocity profiles in the inlet regions of channels or pipes. Such entry flow problems have been investigated extensively, usually by approximate techniques. In a recent paper [4], Horgan & Wheeler have provided an alternative approach, based on an energy method for the stationary Navier-Stokes equations. In [4], concerned with laminar flow in a cylindrical pipe of arbitrary cross-section, an analogy is drawn between the end effect issue of concern here, called the end effect, and the celebrated Saint-Venant's Principle of the theory of elasticity.In this paper, I consider the two-dimensional analog of the problem treated in [4] with a view to providing a more explicit formulation of the energy approach to entry flow problems. The flow development in a semi-infinite channel with parallel-plates is analyzed within the framework of the stationary Navier-Stokes equations. Introduction of a stream function leads to a formulation in terms of a boundary-value problem for a single fourth order nonlinear elliptic equation. In the case of Stokes flow, this problem is formally equivalent to a boundary-value problem for the biharmonic equation considered by Knowles [5] in the analysis of Saint-Venant's Principle in plane elasticity. The main result is an explicit estimate which establishes the exponential spatial flow development and leads to an upper bound for an appropriately defined entrance length. These results are obtained using differential inequality techniques analogous to those developed in investigation of Saint-Venant's Principle.     

On the cross flow through in-line tube banks with regard to the effect of surface roughness

The behaviour of the flow through an in-line tube bank is studied at variable roughness conditions of the surface. The total pressure loss, the local static pressure and the skin friction distribution were measured in the range of Reynolds numbers 4·10⁴<Re<10⁷. Opposing to the staggered arrangement the drag coefficient of an in-line bundle diminishes in the transcritical flow range with increasing roughness parameters. As at the same time heat transfer is improved the in-line bundle is of great practical interest. By means of the local flow parameters the interaction of the point of impact, separation point of the boundary layer and wake width can be pointed out.

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Zusammenfassung Die Strömung durch ein fluchtendes Rohrbündel glatter und rauher Oberfläche wird untersucht. Der Gesamtdruckverlust sowie die örtliche Druck- und Wandschubspannungsverteilung wurden im Bereich der ReynoldsZahlen 4·10⁴<Re<10⁷ gemessen. Im Gegensatz zum versetzten Wärmetauscher verringert sich bei der fluchtenden Anordnung im transkritischen Strömungsbereich der Widerstandsbeiwert mit zunehmender Oberflächenrauhigkeit. Da gleichzeitig eine Verbesserung des Wärmeüberganges beobachtet wird, kommt dem fluchtenden Bündel große praktische Bedeutung zu. Anhand der experimentell gewonnenen örtlichen Strömungsparameter wird die Wechselwirkung von Auftreffpunkt der Strömung, Grenzschichtablösung und Nachlauf breite verdeutlicht.

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Nomenclature D m tube diameter - k _s m sand grain roughness - L m tube length - p N/m₂ pressure - s ₁ m longitudinal pitch - s _tm transversal pitch - U _c m/s velocity in the smallest cross section between the tubes - z number of rows Greek symbols pressure drop coefficient (index t=total, index f=friction) - kg/sm fluid viscosity - kg/m³ fluid density - ₀ N/m² wall shear stresses - grd angle of circumference Characteristics Reynolds number - pressure drop coefficient The investigations have been carried out in the Institut für Reaktorbauelemente der Kernforschungsanlage Jülich GmbH The author wishes to thank Dr.von der Decken, new director of the institute, who supported the already started work in its final stages with great interest. He also expresses his gratitude to his co-workers H.Gillessen, F.Hoffmanns, H.Reger, W.Schmidt and G.Türk for their activity and engagement to bring these investigations to a successful end.     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

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An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

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Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

Experimentelle Bestimmung des Staupunktswärmeüberganges in Argon im Stoßwellenrohr

Zusammenfassung Es wird die Theorie der elektrisch isolierten Kalorimetersonde mit endlicher Schichtdicke entwickelt. Anhand des Ergebnisses lassen sich die möglichen Fehler der Sonde leicht abschätzen. Außerdem wird der Einfluß der Temperaturverteilung auf den Widerstand diskutiert. Die experimentellen Ergebnisse des Wärmeüberganges im Staupunkt einer Halbkugel in Argon und Luft werden befriedigend mit der Theorie vonFay undRiddell erklärt. Bisherige experimentelle Ergebnisse lagen entweder zu hoch oder zu niedrig.

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Experimental determination of the stagnation point heat transfer in argon in a shock tube

The theory of the electrically insulated calorimeter gauge with finite thickness has been developed. From the theoretical results the possible errors of the gauge can be estimated. The influence of the temperature distribution on the electrical resistance is given. The experimental results of the heat transfer to the stagnation point of a spherical cylinder in argon and air agree reasonably with the theory ofFay andRiddell. Earlier experimental results have been found too high or too low.

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Bezeichnungen (Verwendete Einheiten in Klammern) a=/c· Temperaturleitfähigkeit (cm²/s) - c spezifische Wärme (Ws/g grd) - d, L, B Längen (cm) - e=c Wärmeeindringzahl (Ws^1/2/cm² grd) - F Pläche (cm²) - I Strom (A) - M _s Stoß-Machzahl - q Wärmestromdichte (W/cm²) - R Widerstand () - t Zeit (s) - T Temperatur (°K, grd) - U Spannung (V) - x, y, z Raumkoordinaten (cm) - Temperaturkoeffizient des Widerstandes (grd^–1) - Wärmeleitfähigkeit (W/cm grd) - Dichte (g/cm³)     

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.     

Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems

We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension

An integrated multiphase flow sensor for microchannels

The flow regimes of microscale multiphase flows affect the yield and selectivity of microchemical systems, and the heat transfer properties of micro heat exchangers. We describe an integrated optical sensor that uses total internal reflection to detect the structure of multiphase flows in microchannels. The non-intrusive sensor enables detection of individual slugs, bubbles, or drops, and can be used to continuously determine their number and velocity. The sensor performance is modeled using ray-tracing techniques, and tested for several channel geometries. Both gas-liquid and liquid-liquid flows are investigated in microchannels with rectangular and triangular cross-sections. Statistical properties of the flow, derived from the sensor signal, compare favorably to commonly-used dynamic pressure measurements. We demonstrate the integration of the sensor into a planar multichannel microreactor. An existing glass layer used as a waveguide allows us to monitor flows in optically inaccessible channels. This sensor configuration can be integrated into layers of vertically-stacked multichannel microreactors.

List of symbols

Roman symbols a Radius of largest sphere inscribed in channel [m] - A_ch Channel cross-sectional area [m²] - Ca Capillary number [-] - Critical capillary number [-] - d_h Hydraulic diameter [m] - d_sensor Distance prism surface-sensor origin [m] - E₀ Incident light energy [J] - E_r Emerging light energy [J] - f(t_pass) Probability density function (PDF) of slug dwell times [1/s] - f Focal length [m] - f_slug Slug frequency [Hz] - F(t_pass) Probability distribution of slug dwell times [-] - g(t) Arbitrary function of time [-] - h Liquid film thickness [m] - j_G Superficial gas velocity [m/s] - j_L Superficial liquid velocity [m/s] - l Slug length [m] - N Number of samples [-] - n Refractive index [-] - N_c Number of channel corners [-] - n_i Refractive index of incident medium [-] - n_r Number of reflections [-] - n_t Refractive index of transmitting medium [-] - n_slug Number of slugs [-] - p Gas inlet pressure [Pa] - r Reflectance [-] - R_XX(x,) Autocorrelation function [-] - R_Xp(x,) Cross correlation function [-] - r Slug radius at infinite distance from leading slug tip [m] - s Standard deviation of measured slug dwell times [s] - t Time [s] - t Measurement time interval [s] - t_pass Slug dwell time [s] - U_b Slug (bubble) velocity [m/s] - W Bin size of slug dwell time histogram [-] - x Streamwise coordinate [m] - X(x,t) Phase density function [-] - Y Surface tension of the gas-liquid interface [N/m] - Volumetric gas flow rate [m³/s] - Volumetric liquid flow rate [m³/s] - Volumetric oil flow rate [m³/s] - Volumetric water flow rate [m³/s] - z Normal coordinate [m]Greek symbols Void fraction [-] - _c Critical angle for total internal reflection [^°] - _i Incident angle [^°] - Laser wavelength [m] - µ Liquid viscosity [Pa s] - Normalization factor [-] - _h Dimensionless liquid film thickness [-] - _r Dimensionless radius [-] - _x Dimensionless streamwise position [-] - _r Dimensionless slug radius at infinite distance from leading slug end [-] - Standard deviation of the slug dwell time distribution [s] - Time shift [s] - Contact angle [^°]