Heat conduction in a plane wall subject to periodic temperature changes

Analytical solutions for the heat conduction in a plane wall with periodic temperature variations at the wall surface are presented. Series and asymptotic developments of these solutions are deduced. The results are important for the calculation of the heat transfer in rotary kilns or other rotaring units.

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Die Wärmeleitung in einer ebenen Wand mit periodischen Temperaturänderungen

Zusammenfassung Es werden analytische Lösungen für die Wärmeleitung in einer ebenen Wand mit periodischen Temperaturänderungen an ihrer Oberfläche mitgeteilt. Reihen- und asymptotische Entwicklungen dieser Lösungen werden abgeleitet. Die Ergebnisse sind wichtig für die Berechnung des Wärmetransportes in Drehrohröfen oder ähnlichen Maschinen.

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Nomenclature a ² =/ C m²/s thermal diffusivity, Eq. (1) - C J/kg K specific heat - F K initial temperature of the wall, Eq. (4) - F m² surface of the wall - G Green's function, Eq. (10) - G₁ Green's function, Eq. (12) - h m thickness of the wall - H Heaviside function, Eq. (5) - k constant, Eq. (25) - k _x constant, Eq. (25) - Q J total energy, Eq. (17) - Q ^u J total energy from temperatureU, Eq. (18) - Q ^v J total energy from temperatureV, Eq. (19) - s s^–1 Laplace variable - t s time - t ₁ s heating time, Eq. (5) - t ₂ s period, Eq. (5) - T K temperature of the wall - T _i K surface temperature of the wall - T ₁ K surface temperature of the wall during the heating time - T ₂ K surface temperature of the wall during the cooling time - U K temperature of the wall defined in problem 1 - V, K temperature of the wall defined in problem 2 - x m coordinate - ₀ W/m²K overall heat transfer coefficient, Eq. (31) - ₁₀ W/m² K overall heat transfer coefficient, Eq. (32) - ₂₀ W/m² K overall heat transfer coefficient, Eq. (33) - Dirac Delta function - s^–1/2 parameter, Eq. (6) - W/mK thermal conductivity - kg/m³ specific mass - dimensionless time, Eq. (34) - Riemann Zeta function surfix Laplace transformed variable     

Prediction of stagnation point heat transfer for a single round jet impinging on a concave hemispherical surface

This paper deals with a systematic procedure for assessment of fluid flow and heat transfer parameters for a single round jet impinging on a concave hemispherical surface. Based on Scholkemeier's modifications of the Karman-Pohlhausen integral method, expressions are derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. This is followed by the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation is presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Bestimmung des Staupunktes bei der Wärmeübertragung für einen einzelnen Strahl, der auf eine konkave halbkugelige Oberfläche trifft

Zusammenfassung Diese Arbeit beschäftigt sich mit dem systematischen Verfahren der Bewertung von Fluidströmungen und Wärmeübertragungsparametern für einen einzelnen runden Strahl, der auf eine konkave halbkugelförmige Oberfläche trifft. Das Verfahren beruht auf Scholkemeiers Modifikation des Karman-Pohlhausen Integrationsverfahrens. Ausdrücke sind für die Berechnung der Impuls-Dicke, der Grenzschichtdicke und der Verschiebungsdicke am Staupunkt hergeleitet worden. Dies ist aus der Berechnung der thermischen Grenzschichtdicke und des lokalen Wärmeübertragungskoeffizienten abgeleitet worden. Es wird eine Gleichung für die Nusselt-Zahl am Staupunkt als Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände vom Strahlaustrittspunkt bis zum Auftreffpunkt auf die Oberfläche vorgestellt.

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Nomenclature c _p specific heat at constant pressure - d diameter of single round nozzle - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _0.5 Nusselt number based on impinging jet quantities=h _0.50/k - Nu _{0.5, 0} stagnation point Nusselt number=h _{0 0,50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - R radius of curvature of the hemisphere - Re _J jet Reynolds number=U _Jd/ - Re _0.5 Reynolds number based on impinging jet quantities=u _{m0 0.50}/ - T temperature - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions (Fig. 1) - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty= _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - x* non-dimensional coordinate starting at the stagnation point=x/2 _0.50 - x, y rectangular Cartesian coordinates - y coordinate normal to the wall starting at the wall - ratio of thermal to velocity boundary layer thickness= _T/_m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _0.50 jet half width at the plane of impingement if the plate were not there - _m inner boundary layer thickness atu=u _m - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity=/ - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

On the universality of the velocity profiles of a turbulent flow in an axially rotating pipe

If a fluid enters an axially rotating pipe, it receives a tangential component of velocity from the moving wall, and the flow pattern change according to the rotational speed. A flow relaminarization is set up by an increase in the rotational speed of the pipe. It will be shown that the tangential- and the axial velocity distribution adopt a quite universal shape in the case of fully developed flow for a fixed value of a new defined rotation parameter. By taking into account the universal character of the velocity profiles, a formula is derived for describing the velocity distribution in an axially rotating pipe. The resulting velocity profiles are compared with measurements of Reich [10] and generally good agreement is found.Nomenclature b constant, equation (34) - D pipe diameter - l mixing length - l ₀ mixing length in a non-rotating pipe - N rotation rate,N=Re /Re _D - p pressure - R pipe radius - Re _D flow-rate Reynolds number, - Re rotational Reynolds number, Re =v _w D/ - Re* Reynolds number based on the friction velocity, Re*=v*R/ - (Re*)₀ Reynolds number based on the friction velocity in a non-rotating pipe - Ri Richardson number, equation (10) - r coordinate in radial direction - dimensionless coordinate in radial direction, - v _r ,v ,v _z time mean velocity components - v _r ,v ,v _z velocity fluctations - v _w tangential velocity of the pipe wall - v* friction velocity, - axial mean velocity - v _ZM maximum axial velocity - dimensionless radial distance from pipe wall, - y ⁺ dimensionless radial distance from pipe wall - y ₁ ⁺ constant - Z rotation parameter,Z =v _w/v _* =N Re _D /2Re* - _m eddy viscosity - ( _m )₀ eddy viscosity in a non-rotating pipe - coefficient of friction loss - von Karman constant - ₁ constant, equation (31) - density - dynamic viscosity - kinematic viscosity     

Prediction of stagnation point heat transfer for a slot jet impinging on a concave semicylindrical surface

A systematic procedure has been laid out for assessment of fluid flow and heat transfer parameters for a slot jet impinging on a concave semicylindrical surface. Based on Walz's modifications of the Karman-Pohlhausen integral method, expressions have been derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. The work then has been extended for the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation has been presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Berechnung des Wärmeübergangs im Staupunkt eines Strahles, der aus einer rechteckigen öffnung auf eine konkave halbzylindrische Fläche auftrifft

Zusammenfassung Es wurde eine systematische Prozedur für die Abschätzung von Strömungs- und Wärmeübergangsparametern für einen Strahl, der auf eine konkave halbzylindrische Fläche auftrifft, aufgestellt. Basierend auf Walz's Modifikationen der Karman-Pohlhausen Integral-Methode, wurden Ausdrücke für die Berechnung der Impulsdicke, der Grenzschichtdicke und die Versetzungsdicke am Staupunkt abgeleitet. Die Arbeit wurde dann auf die Abschätzung der thermischen Grenzschichtdicke und der lokalen Wärmeübertragungskoeffizienten ausgedehnt. Es wird eine Beziehung für die Nusselt-Zahl am Staupunkt als eine Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände von der Austrittsfläche des Schlitzes bis zur Aufprallfläche aufgestellt.

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Nomenclature c _p specific heat at constant pressure - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _.5 Nusselt number based on impinging jet quantities =h _0.50/k - Nu _.5,0 stagnation point Nusselt number =h _{0 0.50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - p(x*) static pressure at nondimensional distancex* from the stagnation point - Re _J jet Reynolds number =U _J W/ - Re _0.5 Reynolds number based on impinging jet quantities =u _{m0 0.50}/ - T temperature - T* nondimensional temperature =(T–T _W)/(T _J–T _W) - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty = _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - W jet nozzle width - x* nondimensional coordinate starting at the stagnation point =x/2 _0.50 - x, y rectangular cartesian coordinates - y coordinate normal to the wall and starting at the wall - ratio of thermal to velocity boundary layer thickness = _T/ _m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _.50 jet half width at the plane of impingement if the plate were not there - d_.5 free jet (half width) thickness whereu=u _m/2 - _m inner boundary layer thickness atu =u _m - _T thermal boundary layer thickness - nondimensional coordinate normal to wall =y/ _m - _T nondimensional coordinate normal to wall =y/ _T - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity = / - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

Heat transfer in laminar flow in a uniformly porous channel with an applied transverse magnetic field

Summary The steady laminar flow of an incompressible, viscous, and electrically conducting fluid between two parallel porous plates with equal permeability has been discussed by Terrill and Shrestha [6]. In this paper, using the solution of [6] for the velocity field, the heat transfer problems of (i) uniform wall temperature and (ii) uniform heat flux at wall are solved.For small suction Reynolds numbers we find that the Nusselt number, with increasing Reynolds number, increases for case (i) and decreases for (ii).Nomenclature stream function - 2h channel width - x, y distances measured parallel, perpendicular to the channel walls - U velocity of fluid in the x direction at x=0 - V constant velocity of suction at the wall - nondimensional distance, y/h - nondimensional distance, x/h - f() function defined in (1) - density - coefficient of kinematic viscosity - R suction Reynolds number, V h/ - Re channel Reynolds number, 4U h/ - B ₀ magnetic induction - electrical conductivity - M Hartmann number, B ₀ h(/)^1/2 - K constant defined in (3) - A constant defined in (5) - 4R/Re - q local heat flux per unit area at the wall - k thermal conductivity - T temperature of the fluid - X –1/ ln(1–) - C _p specific heat at constant pressure - j current density - Pr Prandtl number, C _p/k - P mass transfer Péclet number, R Pr - Pe mass transfer Péclet number, P/ - T ₀ temperature at x=0 - T _H() temperature in the fully developed region - T _h(X, ) temperature in the entrance region - Y _n () eigenfunctions, uniform wall temperature - _n eigenvalues - e() function defined by (24) - B _n 2/3 _n ² - A _n constants defined by (28) - a _2m constants defined by (30) - F _n () eigenfunctions, uniform wall heat flux - a _n , b _n , c _n , d _n , e _n constants defined by (45) and (48) - S a parameter, U ²/q - h ₁ heat transfer coefficient - T _m mean temperature - Nu Nusselt number - Nu _T Nusselt number, uniform wall temperature - Nu _q Nusselt number, uniform wall heat flux     

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)     

Helium jets discharging normally into a swirling air flow

The characteristics of helium jets injected normally to a swirling air flow are investigated experimentally using laser Doppler and hot-wire anemometers. Two jets with jet-to-crossflow momentum flux ratios of 0.28 and 12.6 are examined. The jets follow a spiral path similar to that found in the swirling air flow alone. Swirl acts to decrease jet penetration, but this is being counteracted by the lighter jet fluid density which is being pressed towards the tube center by the inward pressure gradient. Consequently, in spite of the large variation in momentum flux ratio, jet penetration into the main flow for the two jets investigated is about the same. The presence of the jet is felt only along the spiral path and none at all outside this region. Upstream of the jet, the oncoming swirling flow is essentially unaffected. These characteristics are quite different from jets discharging into a uniform crossflow at about the same momentum flux ratios, and can be attributed to the combined effects of swirl and density difference between the jet fluid and the air stream. Finally, the jets lose their identity in about fifteen jet diameters.List of symbols C mean volume concentration of helium - C _j mean volume concentration of helium at jet exit - c fluctuating volume concentration of helium - instantaneous volume concentration of helium - c RMS volume concentration of helium - D _j jet nozzle diameter - D _T diameter of tube - F flatness factor of c - J = _j U _j ² / _a U _a ^{gn 2} jet-to-crossflow momentum flux ratio - P(c) probability density function of c - r radial coordinate measured from tube centerline - R = D _T /2 radius of tube - Re _j = D _j U _j / _j jet Reynolds number - S = = tan swirl number - Sk skewness of c - instantaneous axial velocity - u RMS axial velocity - U mean axial velocity - local average mean axial velocity across tube - U _j jet exit velocity - U _a overall average mean axial velocity across tube - instantaneous circumferential velocity - w RMS circumferential velocity - W mean circumferential velocity - x axial coordinate measured from exit plane of swirler - x ₁ axial coordinate measured from centerplane of normal jet - y normal distance measured from tube wall - _j jet fluid kinematic viscosity - _a air density - _j jet fluid density - vane angle (constant)     

Interaction between thermocapillary and buoyancy driven convection

Convective flows driven by the variation of surface tension due to a radial temperature gradient along a liquid-gas interface were studied. Three liquids of different viscosities were applied, so that a wide range of Marangoni numbers was encountered. Light sheet technique and differential interferometry were taken to analyse the thermal flows. The mechanism of stationary thermocapillary convection, the influence of the radial temperature gradient and the kinematic viscosity on the Marangoni boundary layer thickness are discussed. Transitions from the steady to the oscillatory Marangoni convection are discovered and the oscillations are visualized with differential interferometry.List of symbols a thermal diffusivity - D cell diameter - f tangential stress - H cell height - Mg Marangoni number, Mg = U · R/a - Pr Prandtl number, Pr = v/a - r radial coordinate tangential to the interface - R cell radius - Re Reynolds number, Re = UR/v - T temperature - T _b, T_m temperature at the boundary and in the centre of the cell, respectively - T temperature difference, T — T _b — T_m - U reference velocity, U = ¦d/dT¦(T/R) R/ - v _r radial stream velocity - v _x velocity at the interface - z axial coordinate normal to the interface - dynamic viscosity - kinematic viscosity - surface tension - d/dT thermal coefficient of surface tension A version of this paper was presented at the 7th Physico-Chemical Hydrodynamics, PCH Conference, June 25–29, 1989, Cambridge, MA, USA     

Injection moulding of thermoplastics: Freezing of variable-viscosity fluids.

The injection moulding of thermoplastics involves, during mould filling, flows of hot polymer melts into mould networks, the walls of which are so cold that frozen layers form on them. An analytical study of such flows is presented here for the case when the Graetz and Nahme numbers are large and the Pearson number is small. Thus the flows are developing and temperature differences due to heat generation by viscous dissipation are sufficiently large to cause significant variations in viscosity (but the difference between the entry temperature of the polymer to a specific part of the mould network and the melting temperature of the polymer is not). Br Brinkman number - Gz Graetz number - h half-height of channel or disc - h ^* half-height of polymer melt region in channel or disc - L length of channel or pipe - m viscosity shear-rate exponent - Na Nahme number - p pressure - P pressure drop - Pe Péclet number - Pn Pearson number - Q volumetric flowrate - r radial coordinate in pipe or disc - R radius of pipe - Re Reynolds number - R _i inner radius of disc - R _o outer radius of disc - R ^* radius of polymer melt region in pipe - T temperature - T _ad adiabatic temperature rise - T _e entry polymer melt temperature - T _m melting temperature of polymer - T _max maximum temperature - T ₀ reference temperature - T _w wall temperature - flow-average temperature rise - u _r radial velocity in pipe or disc - u _x axial velocity in channel - u _y transverse velocity in channel or disc - u _z axial velocity in pipe - w width of channel - x axial coordinate in channel or modified radial coordinate in disc - y transverse coordinate in channel or disc - z axial coordinate in pipe - thermal conductivity of molten polymer - thermal conductivity of frozen polymer - scaled dimensionless axial coordinate in channel or pipe or radial coordinate in disc - ₀ undetermined integration constant - heat capacity of molten polymer - viscosity temperature exponent - dimensionless transverse coordinate in channel or disc - ^* dimensionless half-height of polymer melt region in channel or disc - H ^* scaled dimensionless half-height of polymer melt region in channel or disc or radius of polymer melt region in pipe - dimensionless temperature - ^* dimensionless wall temperature - scaled dimensionless temperature - numerical constant - µ viscosity of molten polymer - µ ₀ consistency of molten polymer - dimensionless pressure gradient - scaled dimensionless pressure gradient - density of molten polymer - dimensionless radial coordinate in pipe or disc - _i dimensionless inner radius of disc - ^* dimensionless radius of polymer melt region in pipe - dimensionless streamfunction - scaled dimensionless streamfunction - dummy variable - streamfunction - similarity variable - similarity variable     

Mixing enhancement in axisymmetric turbulent isothermal and buoyant jets

Measurements of the velocity and concentration in axisymmetric, turbulent, isothermal and buoyant jets have been performed with laser-Doppler velocimetry and planar and point laser-induced fluorescence to quantify the mixing enhancement achieved by periodic forcing when the jet exit has a fully-developed turbulent pipe flow, a situation less well-studied than the case of laminar initial conditions. It was found that forcing at Strouhal numbers around 0.6 enhances mixing in the developing region of the jet and this enhancement increased with increasing amplitude of excitation, consistent with results of initially-laminar jets. The initial turbulence intensity did not have any effect, but an increase in the initial lengthscale of the turbulence, controlled by a perforated plate inside the nozzle, caused faster mixing. In agreement with previous experiments, the initial conditions of the jet did not affect the far-field rate of decay, but the jet-fluid concentration there was significantly reduced by forcing due to the increased mixing during the early stages of development, an effect that can be described by a smaller virtual origin in decay laws of jet decay. These results are independent of the Froude number because the initial conditions have an influence only in the early stages where the flow is still momentum dominated.List of Symbols A normalised excitation amplitude, defined by A = u'/U ₀ - D nozzle diameter - f jet-fluid concentration - F mean f - f r.m.s. f - Fd Froude number, defined by Fd=U ₀ ² /(gDT ₀) - g acceleration of gravity - I fluorescent intensity - I _inc incident light intensity - I _ref light intensity of the reference flow - K decay constant - L _hf concentration halfwidth - M mixing enhancement, defined by U _cl/U _cl,st=0 at x/D=5 - r radial coordinate - Re Reynolds number, defined by Re=U ₀ D/v - [Rh] concentration of Rhodamine B - St Strouhal number, defined by St=D/U ₀ - T ₀ temperature of jet fluid - T temperature of outer fluid - T ₀ temperature difference (= T ₀–, T ) - u r.m.s. axial velocity - u r.m.s. of the sinusoidal velocity fluctuation due to forcing - U mean axial velocity - U _cl mean axial centreline velocity - U _cl,st=0 mean axial centreline velocity for an unforced jet - U _max U at the centre of the nozzle exit - U ₀ bulk velocity at nozzle exit - x streamwise coordinate - X ₀ virtual origin Greek coefficient of thermal expansion - kinematic viscosity of the jet fluid - forcing frequency The experiments described here have been performed together with Mr. J. Sakakibara. Acknowledgments are also due to Prof. H. Longmire, of the University of Minnesota, for helpful discussions on forcing. This work was done while E.M. visitied Keio University with the financial assistance of TEPCO.     

Heat transport in a parabolic tube of circular cross-section

The local temperature has been determined for a viscous liquid flowing through a paraboloidal tube. Wall temperature and inlet temperature have been considered constant. The liquid flow was considered as creeping flow and its velocity distribution was determined by solving the biharmonic differential equation of the stream function. The local temperature was evaluated numerically from the analytical results.

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Wärmetransport im Paraboloidrohr

Zusammenfassung Es wird die lokale Temperatur in einer viskosen Strömung durch ein Paraboloidrohr bestimmt. Dabei wird konstante Wand- und Einlauftemperatur angenommen. Die kriechende Strömungsgeschwindigkeit wurde aus der Lösung der biharmonischen Differentialgleichung der Stromfunktion bestimmt. Die lokale Temperatur wurde aus den analytischen Ergebnissen für einige Paraboloidrohre numerisch bestimmt.

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Nomenclature ₁ F ₁ confluent hypergeometric function - diffusivity - T(, , ) temperature - T _w temperature at the paraboloidal wall - T _i temperature at the inlet - u(, ) flow velocity of viscous liquid in -direction - volumetric flow - eigenvalues of confluent hypergeometric function - streamfunction - _o wall of paraboloidal tube - _o inlet of paraboloidal tube - , , paraboloidal coordinates     

Darcy-Forchheimer natural,forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.Nomenclature A cross-sectional area - b _i coefficient in the chosen temperature profile - B ₁ coefficient in the profile for the dimensionless boundary layer thickness - C coefficient in the modified Forchheimer term for power-law fluids - C ₁ coefficient in the Oseen approximation which depends essentially on pore geometry - C _i coefficient depending essentially on pore geometry - C _D drag coefficient - C _t coefficient in the expression forK ^* - d particle diameter (for irregular shaped particles, it is characteristic length for average-size particle) - f _p resistance or drag on a single particle - F _R total resistance to flow offered byN particles in the porous media - g acceleration due to gravity - g _x component of the acceleration due to gravity in thex-direction - Grashof number based on permeability for power-law fluids - K intrinsic permeability of the porous media - K ^* modified permeability of the porous media for flow of power-law fluids - l _c characteristic length - m exponent in the gravity field - n power-law index of the inelastic non-Newtonian fluid - N total number of particles - Nux,_D,F local Nusselt number for Darcy forced convection flow - Nux,_D-F,F local Nusselt number for Darcy-Forchheimer forced convection flow - Nux,_D,M local Nusselt number for Darcy mixed convection flow - Nux,_D-F,M local Nusselt number for Darcy-Forchheimer mixed convection flow - Nux,_D,N local Nusselt number for Darcy natural convection flow - Nux,_D-F,N local Nusselt number for Darcy-Forchheimer natural convection flow - pressure - p exponent in the wall temperature variation - _Pe c characteristic Péclet number - _Pe x local Péclet number for forced convection flow - _Pe x modified local Péclet number for mixed convection flow - _Ra c characteristic Rayleigh number - _Ra x local Rayleigh number for Darcy natural convection flow - _Ra x local Rayleigh number for Darcy-Forchheimer natural convection flow - Re convectional Reynolds number for power-law fluids - Reynolds number based on permeability for power-law fluids - T temperature - T _e ambient constant temperature - T w,_ref constant reference wall surface temperature - T _w(X) variable wall surface temperature - T _w temperature difference equal toT w,_ref–T _e - T ₁ term in the Darcy-Forchheimer natural convection regime for Newtonian fluids - T ₂ term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation) - T _N term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation) - u Darcian or superficial velocity - u ₁ dimensionless velocity profile - u _e external forced convection flow velocity - u _s seepage velocity (local average velocity of flow around the particle) - u _w wall slip velocity - U c _M characteristic velocity for mixed convection - U c _N characteristic velocity for natural convection - x, y boundary-layer coordinates - x ₁,y ₁ dimensionless boundary layer coordinates - X coefficient which is a function of flow behaviour indexn for power-law fluids - effective thermal diffusivity of the porous medium - shape factor which takes a value of/4 for spheres - shape factor which takes a value of/6 for spheres - ₀ expansion coefficient of the fluid - _T boundary-layer thickness - T ₁ dimensionless boundary layer thickness - porosity of the medium - similarity variable - dimensionless temperature difference - coefficient which is a function of the geometry of the porous media (it takes a value of 3 for a single sphere in an infinite fluid) - ₀ viscosity of Newtonian fluid - ^* fluid consistency of the inelastic non-Newtonian power-law fluid - constant equal toX(2 ^2–n )/ - density of the fluid - dimensionless wall temperature difference     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Evaluation of the performance characteristics of a thermal transient anemometer

The Goertler instability of a hypersonic boundary layer and its influence on the wall heat transfer are experimentally analyzed. Measurements, made in a wind tunnel by means of a computerized infrared (IR) imaging system, refer to the flow over two-dimensional concave walls. Wall temperature maps (that are interpreted as surface flow visualizations) and spanwise heat transfer fluctuations are presented. Measured vortices wavelengths are correlated to non-dimensional parameters and compared with numerical predictions from the literature.List of symbols c _p Specific heat coefficient at constant pressure of the free stream - F Input (true) image - F ₀ Fourier number - Restored image - G Recorded (degraded) image - G Goertler number based on the boundary layer thickness, as defined by Eq. (3) - H System transfer function - M Mach number - Pr Prandtl number - p ₀ Stagnation pressure - Exchanged net heat flux - Convective heat flux - Radiative heat flux - r Recovery factor - Re _m Unit Reynolds number - Re _x Local Reynolds number based on the distance from the leading edge - Re Local Reynolds number based on the boundary layer thickness - Curvature radius - St Stanton number, as defined by Eq. (7) - T _aw Adiabatic wall temperature - T _w Wall temperature - T ₀ Stagnation temperature - t Time - V Free stream velocity - x Streamwise spatial coordinate - y Normal-to-wall spatial coordinate - z Spanwise spatial coordinate - Thermal diffusivity coefficient - Disturbance wavenumber - Non dimensional wavenumber - Boundary layer thickness - Goertler number based on the vortices wavelength - Vortices wavelength - Free stream density - Disturbance total amplification, as defined by Eq. (3) - Disturbance (spatial) growth rate - Non-dimensional growth rate - Perturbation amplitude of a generic quantity - Perturbation amount     

Flow of second-order fluids over an enclosed rotating disc

Summary This paper is devoted to a study of the flow of a second-order fluid (flowing with a small mass rate of symmetrical radial outflow m, taken negative for a net radial inflow) over a finite rotating disc enclosed within a coaxial cylinderical casing. The effects of the second-order terms are observed to depend upon two dimensionless parameters ₁ and ₂. Maximum values ₁ and ₂ of the dimensionless radial distances at which there is no recirculation, for the cases of net radial outflow (m>0) and net radial inflow (m<0) respectively, decrease with an increase in the second-order effects [represented by T(=₁+₂)]. The velocities at ₁ and ₂ as well as at some other fixed radii have been calculated for different T and the associated phenomena of no-recirculation/recirculation discussed. The change in flow phenomena due to a reversal of the direction of net radial flow has also been studied. The moment on the rotating disc increases with T.Nomenclature , , z coordinates in a cylindrical polar system - z ₀ distance between rotor and stator (gap length) - =/z ₀, dimensionless radial distance - =z/z ₀, dimensionless axial distance - _s = _s/z₀, dimensionless disc radius - V =(u, v, w), velocity vector - dimensionless velocity components - uniform angular velocity of the rotor - , p fluid density and pressure - P =p/(₂ z ₀₂ ² , dimensionless pressure - ₁, ₂, ₃ kinematic coefficients of Newtonian viscosity, elastico-viscosity and cross-viscosity respectively - ₁, _{2 2}/z ₀ ² , resp. ₃/z ₀ ² , dimensionless parameters representing the ratio of second-order and inertial effects - m = , mass rate of symmetrical radial outflow - l a number associated with induced circulatory flow - Rm =m/(z ₀₁), Reynolds number of radial outflow - R _l =l/(z ₀₁), Reynolds number of induced circulatory flow - Rz =z ₀ ² /₁, Reynolds number based on the gap - ₁, ₂ maximum radii at which there is no recirculation for the cases Rm>0 and Rm<0 respectively - ₁(T), ₂(T) ₁ and ₂ for different T - U _1(T) ⁽⁺⁾ = dimensionless radial velocity, Rm>0 - V _1(T) ⁽⁺⁾ = , dimensionless transverse velocity, Rm>0 - U _2(T) ^(–) = , dimensionless radial velocity, Rm=–Rn<0, m=–n - V _2(T) ^(–) = , dimensionless transverse velocity, Rm<0 - C _m moment coefficient     

Measurements of thermally stratified pipe flow using image-processing techniques

The cross-correlation technique and Laser Induced Fluorescence (LIF) have been adopted to measure the time-dependent and two-dimensional velocity and temperature fields of a stably thermal-stratified pipe flow. One thousand instantaneous and simultaneous velocity and temperature maps were obtained at overall Richardson numberRi = 0 and 2.5, from which two-dimensional vorticity, Reynolds stress and turbulent heat flux vector were evaluated. The quasi-periodic inclined vortices (which connected to the crest) were revealed from successive instantaneous maps and temporal variation of vorticity and temperature. It has been recognized that these vortices are associated with the crest and valley in the roll-up motion.List of symbols A Fraction of the available light collected - C Concentration of fluorescence - D Pipe diameter - I Fluorescence intensity - L Sampling length along the incident beam - I ₀ Intensity of an excitation beam - I _c (T) Calibration curve between temperature and fluorescence intensity - I _ref Reference intensity of fluorescence radiation - Re _b Reynolds number based on bulk velocity,U _b D/v - Ri Overall Richardson number based on velocity difference,gDT/U ² - t Time - t Time interval between the reference and corresponding matrix - T Temperature - T ₁,T ₂ Temperature of lower and upper layer - T ^* Normalized temperature, (T–T ₁)/T - T _c (I) Inverse function of temperature as a function ofI _c - T _ref Reference temperature - T Temperature difference between upper and lower flow,T ₂ –T ₁ - U ₁ Velocity of lower stream - U ₂ Velocity of upper stream - U _b Bulk velocity - U _c Streamwise mean velocity atY/D=0 - U Streamwise velocity difference between upper and lower flow,U ₁–U ₂ - u, v, T Fluctuating component ofU, V, T - U, V Velocity component of X, Y direction - X Streamwise distance from the splitter plate - Y Transverse distance from the centerline of the pipe - Z Spanwise distance from the centerline of the pipe - Quantum yield - Absorptivity - vorticity calculated from a circulation - Kinematic viscosity - circulation     

Oscillatory free convection heat transfer along a semi-infinite vertical plate

Summary The fluctuating free convection flow along a semi-infinite vertical plate is considered when the plate temperature is of the form T _p –T =(T ₀ –T ) where 0 < 1, denotes the frequency of oscillation and the mean temperature T ₀–T is proportional to ⁿ (0 n < 1). Flow and temperature fields have been obtained by means of two asymptotic expansions. For small values of the frequency parameter , a regular expansion is obtained while for large the method of matched asymptotic expansion is used. It is found that the skin friction and the rate of heat transfer obtained from two expansions overlap satisfactorily for a certain value of . For n=1 the flow governing equations to a semisimilar form, and have been solved by finite difference method. The results obtained from the series and the finite difference methods are in good agreement.

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Oszillierender Wärmeübergang an einer halbunendlichen senkrechten Platte bei freier Konvektion

Übersicht Betrachtet wird die fluktuierende freie Konvektionsströmung längs einer halbunendlichen senkrechten Platte, deren Temperatur dem Gesetz T _p –T =(T ₀–T ) [1+ sin {ie1-03}] folgt, wobei 0 < 1 gelte, {ie1-04} die Frequenz ist und der Temperatur-Mittelwert T ₀–T proportional zu ⁿ (0 n < 1) ist. Mit Hilfe zweier asymptotischer Entwicklungen werden die Strömungs- und Temperaturfelder gewonnen. Für kleine Werte des Frequenzparameters wird eine gewöhnliche Entwicklung benutzt, für große die Methode angepaßter asymptotischer Entwicklungen. Es stellt sich heraus, daß die Oberflächenreibung und die Wärmeübergangsrate aus zwei Entwicklungen für ein bestimmtes zufriedenstellend aufeinander fallen. Für n=1 werden die Grundgleichungen zueinander ähnlich und werden nach der Finite-Differenzen-Methode gelöst. Die Ergebnisse nach den Reihenentwicklungen und der Finite-Differenzen-Methode stimmen gut überein.

     

Heat transfer between gas-liquid spray stream flowing perpendicularly to the row of the cylinders

Experimental investigation and analysis of heat transfer process between a gas-liquid spray flow and the row of smooth cylinders placed in the surface perpendicular to the flow has been performed. Among others, there was taken into account in the analysis the phenomenon of droplets bouncing and omitting the cylinder as well as the phenomenon of the evaporation process from the liquid film surface.In the experiments test cylinders were used, which were placed between two other cylinders standing in the row.From the experiments and the analysis the conclusion can be drawn that the heat transfer coefficients values for a row of the cylinders are higher than for a single cylinder placed in the gasliquid spray flow.

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Wärmeübergang an eine senkrecht anf eine Zylinderreihe auftreffende Gas-Flüssigkeits-Sprüh-Strömung

Zusammenfassung Es wurden Messungen und theoretische Analysen des Wärmeübergangs zwischen einer Gas-FlüssigkeitsSprüh-Strömung und den glatten Oberflächen einer Zylinderreihe durchgeführt, die senkrecht zum Sprühstrahl angeordnet waren. Dabei wurde in der Analyse unter anderem das Phänomen betrachtet, daß die Tropfen die Zylinderwand treffen und verfehlen können und daß sich ein Verdampfungsprozeß aus dem flüssigen Film an der Zylinderoberfläche einstellt.Gemessen wurde an einem zwischen zwei Randzylindern befindlichen Zylinder.Die Experimente und die Analyse gestatten die Schlußfolgerung, daß der Wärmeübergangskoeffizient für eine Zylinderreihe höher ist als für einen einzelnen Zylinder in der Sprühströmung.

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Nomenclature a distance between axes of cylinders, m - c _l specific heat capacity of liquid, J/kg K - c _g specific heat capacity of gas, J/kg K - D cylinder diameter, m - g _l mass velocity of liquid, kg/m²s - ¯k average volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - k() local volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - L specific latent heat of vaporisation, J/kg - m mass fraction of water in gas-liquid spray flow, dimensionless - M constant in Eq. (9) - p pressure, Pa - p _g statical pressure of gas, Pa - p _w pressure of gas on the cylinder surface, Pa - p external pressure on the liquid film surface, Pa - r cylindrical coordinate, m - R radius of cylinder, m - T temperature, K, °C - T _l, t_l liquid temperature in the gas-liquid spray, K, °C - T _w,t_w temperature of cylinder surface, K, °C - T temperature of gas-liquid film interface, K - U liquid film velocity, m/s - w gas velocity on cylinder surface, m/s - w _g gas velocity in free stream, m/s - W _l liquid vapour mass ratio in free stream, dimensionless - W liquid vapour mass ratio at the edge of a liquid film, dimensionless - x coordinate, m - y coordinate, m - z complex variable, dimensionless - average heat transfer coefficient, W/m²K - local heat transfer coefficient, W/m² K - average heat transfer coefficient between cylinder surface and gas, W/m² K - _g, local heat transfer coefficient between cylinder surface and gas, W/m² K - mass transfer coefficient, kg/m²s - liquid film thickness, m - _lg dynamic diffusion coefficient of liquid vapour in gas, kg/m s - pressure distribution function on a cylinder surface - function defined by Eq. (3) - _l liquid dynamic viscosity, kg/m s - _g gas dynamic viscosity, kg/m s - cylindrical coordinate, rad, deg - _l thermal conductivity of liquid, W/m K - _g thermal conductivity of gas, W/m K - mass transfer driving force, dimensionless - _l density of liquid, kg/m³ - _g density of gas, kg/m³ - _w shear stress on the cylinder surface, N/m² - _w shear stress exerted by gas at the liquid film surface, N/m² - air relative humidity, dimensionless - T -T _w - _w =T _wT_l Dimensionless parameters I= enhancement factor of heat transfer - m ^*=M _l/M_g molar mass of liquid to the molar mass of gas ratio - Nu _g= D/ _g gas Nusselt number - Pr _g=c _{g g}/_g gas Prandtl number - Pr _l=c_lll liquid Prandtl number - ¯r=(r–R)/ dimensionless coordinate - Re _g=w_gD _g/_g gas Reynolds number - Re _g,max=w_g,max D _g/_g gas Reynolds number calculated for the maximal gas velocity between the cylinders - Sc=m ^* _g/_l–g Schmidt number =/R dimensionless film thickness     

Low power laser Rayleigh probe for flow mixing studies

A laser Rayleigh correlation probe was constructed, which allows the application of low cost, low power (milliwatt) laser sources. It was tested for basic mixing studies in isothermal binary gas flows. Here, it can be used for the time and space resolved measurement of the concentration mean value and of all important statistical quantities, which give information on the distribution around the concentration mean value (rms, skewness, kurtosis) and on the relation of adjecent fluctuations in time or space (autocorrelation function, power spectral density).List of symbols c concentration (mole fraction) of investigated gas species - c time averagered mean concentration - c instantaneous fluctuating concentration - rms concentration - D Rayleigh intensity difference of two gas species (I _R1–I _R2) - d width of the rectangular channels (x-direction), see Fig. 3 - f frequency - G() Rayleigh autocorrelation function (ACF) - I ₀ intensity of irradiated laser light - I _Ri intensity of Rayleigh signal of gas species i - K, k calibration constant of Rayleigh probe - l lenght of observed scattering volume - n(t) temporally fluctuating number density of gas molecules - R() normalized ACF - S Rayleigh intensity of gas components 2 in a binary mixture (I _R2) - T gas temperature - t time - u exit velocity - skewness of the concentration distribution around the mean value - kurtosis of the concentration distribution around the mean value - (d/d)_eff effective scattering cross section of the binary gas mixture - solid angle of collection optics - delay time - sample time     

Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid

We study the modelling of purely conductive heat transfer between a porous medium and an external fluid within the framework of the volume averaging method. When the temperature field for such a system is classically determined by coupling the macroscopic heat conduction equation in the porous medium domain to the heat conduction equation in the external fluid domain, it is shown that the phase average temperature cannot be predicted without a generally negligible error due to the fact that the boundary conditions at the interface between the two media are specified at the macroscopic level.Afterwards, it is presented an alternative modelling by means of a single equation involving an effective thermal conductivity which is a function of point inside the interfacial region.The theoretical results are illustrated by means of some numerical simulations for a model porous medium. In particular, temperature fields at the microscopic level are presented.Roman Letters _sf interfacial area of thes-f interface contained within the macroscopic system m² - A _sf interfacial area of thes-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - g vector that maps to _s , m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l _s,l _f microscopic characteristic length m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for thef-phase at thef-s interface - n outwardly directed unit normal vector at the dividing surface. - R ₀ REV characteristic length, m - T _i macroscopic temperature at the interface, K - error on the external fluid temperature due to the macroscopic boundary condition, K - T ^* macroscopic temperature field obtained by solving the macroscopic Equation (3), K - V averaging volume, m³ - V _s,V _f volume of the considered phase within the averaging volume, m³. - _mp volume of the porous medium domain, m³ - _ex volume of the external fluid domain, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² - x, z spatial coordinates Greek Letters _s, _f volume fraction - ratio of the effective thermal conductivity to the external fluid thermal conductivity - ^* macroscopic thermal conductivity (single equation model) kcal/m s K - _s, _f microscopic thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding toT ^*, K - spatial deviation temperature K - error in the temperature due to the macroscopic boundary conditions, K - ^* _i macroscopic temperature at the interface given by the single equation model, K - spatial average - ^s , ^f intrinsic phase average.