Vaporization of single liquid drops in an immiscible liquid part II: Heat transfer characteristics

Heat transfer characteristics during the vaporization process of a pentane or furan drop in an aqueous glycerol of high viscosity has been studied. With the progress of vaporization, the overall heat transfer coefficient related to the liquid-liquid interfacial area of a two-phase bubble increases monotonically, and influences of initial drop diameter and temperature difference reduce. Some convection or circulation seems to occur in the unvaporized-liquid phase.

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Verdampfung einzelner Flüssigkeitstropfen in einer nicht mischbaren Flüssigkeit. Teil II: Der Wärmeübergang

Zusammenfassung In dieser Arbeit wird der Wärmeübergang während der Verdampfung von Pentan- und Furan-Tropfen in einer wässerigen Glyzerinlösung hoher Viskosität untersucht. Mit fortschreitender Verdampfung steigt der Wärmeübergangskoeffizient, bezogen auf die Grenzfläche flüssig-flüssig der zweiphasigen Blase monoton an, wobei Einflüsse des anfänglichen Tropfendurchmessers und der Temperaturdifferenz abnehmen. In der nichtverdampften Flüssigkeitsphase scheint Konvektion oder Zirkulation aufzutreten.

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Nomenclature A total surface area of two-phase bubble - A_L liquid-liquid interfacial area of two-phase bubble - D equivalent spherical diamter of two-phase bubble - D_i initial drop diameter - h average overall heat transfer coefficient related to A - h_c average outside heat transfer coefficient related to A - q local outside heat transfer coefficient - h_L average overall heat transfer coefficient related to A_L - h_Lc average outside heat transfer coefficient related to A_L - k_c thermal conductivity of continuous-phase liquid - k_dl thermal conductivity of dispersed-phase liquid - k_v correction factor of velocity [cf. Eq.(2)] - Nu_c =h_c D/k - Nu_c =h_c D/k_c - Pe_c =UD/_c - Pr_c =_c/_c - Q cumulative heat transferred into two-phase bubble - q local heat flux - r radial distance in spherical co-ordinates - R radius of two-phase bubble - T temperature - T_L interface temperature between continuousphase and dispersed-phase component in liquid phase - T bulk temperature - T temperature difference - T nominal temperature difference - U velocity of rise of two-phase bubble - u velocity gradient in r direction [cf. Eq.(9)] - u_r velocity component in r direction - u velocity component in direction - V volume of two-phase bubble - V_dl volume of dispersed-phase component in liquid phase - X defined in Eq.(7) - Y defined in Eq.(8) - Z defined in Eq.(12) - _c thermal diffusivity of continuous-phase liquid - half opening angle of vapor phase in two-phase bubble - average thickness of dispersed-phase component in liquid phase [cf. Eq.(22)] - angle in spherical co-ordinates - vaporization ratio - time     

Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping (1/2 order subharmonic oscillations and entrainment)

Nonlinear forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping are studied. In particular, entrainment phenomena at the critical speeds of 1/2 order subharmonic oscillations of forward and backward whirling modes are investigated. A self-excited oscillation appears in the wide range above the major critical speed. The amplitude of this oscillation reaches a limit value and then a self-sustained oscillation occurs. In the vicinity of a 1/2 order subharmonic oscillation of a forward whirling mode, a self-excited oscillation is entrained by a subharmonic oscillation. In the vicinity of a 1/2 order subharmonic oscillation of a backward whirling mode, either a self-excited oscillation or a subharmonic oscillation occurs.Experiments were made by an elastic rotating shaft with a disc. Nonlinearity in its restoring force was due to an angular clearance of a bearing and internal damping was due to friction between the shaft and an inner ring of the bearing. A self-excited oscillation was observed in the range above the major critical speed and this self-excited oscillation was entrained by a 1/2 order subharmonic oscillation of a forward whirling mode.Nomenclature O–xyz rectangular coordinate system - , _x, _y inclination angle of a shaft and its projections on the xz- and yz-planes - _x, _y inclination angles in rotating coordinates - , polar coordinates - I _p polar moment of inertia of a rotor - I diametral moment of inertia of a rotor - i _p ratio of I _p to I - dynamic unbalance of a rotor - rotating speed (angular velocity) - F magnitude of a dynamic unbalance force, F = (1 – i _p )² - c external damping coefficient - h internal damping coefficient - t time - D _x , D _y internal damping terms in stationary coordinates - D _x , D _y internal damping terms in rotating coordinates - N _x , N _y nonlinear terms in restoring forces     

Injection moulding of thermoplastics: Freezing during mould filling

The injection moulding of thermoplastic polymers involves, during mould filling, flows of hot melts into mould networks, the walls of which are so cold that frozen layers form on them. Theoretical analyses of such flows are presented here. Br Brinkman number - c _L polymer melt specific heat capacity - c _S frozen polymer specific heat capacity - e exponential function - erf() error function - Gz Graetz number in thermal entrance region - Gz ^* modified Graetz number in thermal entrance region - Gz overall Graetz number - h channel half-height - h ^* half-height of polymer melt region - H mean heat transfer coefficient - k _L polymer melt thermal conductivity - k _S frozen polymer thermal conductivity - ln( ) natural logarithm function - L length of thermal entrance region in pipe or channel - m viscosity shear rate exponent - M(,,) Kummer function - Nu Nusselt number - p pressure - P pressure drop in thermal entrance region - P _f pressure drop in melt front region - Pe Péclet number - Pr Prandtl number - Q volumetric flow rate - r radial coordinate in pipe - R pipe radius - R ^* radius of polymer melt region - Re Reynolds number - Sf Stefan number - t time - T temperature - T _i inlet polymer melt temperature - T _m melting temperature of polymer - T _w pipe or channel wall temperature - U(,,) Kummer function - u _r radial velocity in pipe - u _x axial velocity in channel - u _y cross-channel velocity - u _z axial velocity in pipe - V melt front velocity - w channel width - x axial coordinate in channel - x _f melt front position in channel - y cross-channel coordinate - z axial coordinate in pipe - z _f melt front position in pipe - () gamma function - dimensionless thickness of frozen polymer layer - _i i-th term (i = 1,2,3) in power series expansion of - dimensionless axial coordinate in pipe - _f dimensionless melt front position in pipe - dimensionless cross-channel coordinate - ^* dimensionless half-height of polymer melt region - dimensionless temperature - _i i-th term (i = 0, 1, 2, 3) in power series expansion of - _i first derivative of _i with respect toø - _i second derivative of _i with respect toø - ^* dimensionless wall temperature - thermal diffusivity ratio - - latent heat of fusion - µ viscosity - µ ^* unit shear rate viscosity - dimensionless axial coordinate in channel - _f dimensionless melt front position in channel - dimensionless pressure drop in thermal entrance region - _f dimensionless pressure drop in melt front region - _L polymer melt density - _s frozen polymer density - dimensionless radial coordinate in pipe - ^* dimensionless radius of polymer melt region - ø dimensionless similarity variable in thermal entrance region - dummy variable - dimensionless contracted axial coordinate in thermal entrance region - dimensionless similarity variable in melt front region - ^* constant     

Flow about a circular cylinder with a single large-scale surface perturbation

An experimental study of the flow around a cylinder with a single straight perturbation was conducted in a wind tunnel. With this bluff body, positioned in a uniform crossflow, the vortex shedding frequency and other flow characteristics could be manipulated.The Strouhal number has been shown to be a function of the perturbation angular position, _p , as well as the perturbation size and Reynolds number. As much as a 50% change in Strouhal number could be achieved, simply by changing _p by 1°. The perturbation size compared to the local boundary layer thickness, , was varied from approximately 1 to about 20 . The Reynolds number was varied from 10,000 to 40,000. For perturbation sizes approximately 5 to 20 and Reynolds numbers of 20,000 to 40,000, a consistent Strouhal number variation with _p was observed.A detailed investigation of the characteristic Strouhal number variation has shown that varying _p had a significant influence on the boundary layer separation and transition to turbulence. These significant changes occurring in the boundary layer have been shown to cause variations in the spacing between the shear layers, base pressure, drag, lift, and the longitudinal spacing between the vortices in the vortex street.List of Symbols a longitudinal spacing of vortices in the vortex street - C _d drag coefficient - C _dc drag coefficient corrected for blockage effect - C _l lift coefficient - C _p pressure coefficient, p _i –p /q - C _pb base pressure coefficient - C _pbc base pressure coefficient corrected for blockage effect - d perturbation diameter - d ^* spacing between the shear layers; defined as conditionally averaged spacing between points in the shear layers corresponding to 0.99u _max/U - D cylinder diameter; diameter of the circumscribing circle for a cable - f _v vortex shedding frequency - H wind tunnel test section cross-sectional width - L spanwise length of the cylinder - p _i tap pressure - p free stream static pressure - q free stream dynamic pressure - Re Reynolds number based on cylinder diameter - rms root-mean-square - S Strouhal number, f _v D/U - S _max maximum value of S - S _min minimum value of S - t time - u _c vortex convection velocity - u _max maximum velocity in the shear layer - U free stream velocity - U _c free stream velocity, corrected for blockage effect - x streamwise dimension referenced from the back of the cylinder - z lateral wake dimension, i.e., perpendicular to the free stream velocity vector and cylinder axis, referenced from the cylinder axis - x spacing between two hot wire probes aligned streamwise - phase difference between two hot wire probes aligned streamwise - boundary layer thickness - angle from stagnation point in degrees - _p perturbation angular position - _{b p} where S drops back to about the S of a cylinder - _c critical angle, angular position where S drops steeply with 1° change in - _{m p} where S was minimum - _{r p} after S recovers from drop in value - _{t p} where S starts to increase from about the S of a cylinder     

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

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

Laminar thermally developing flow inside right-angularly triangular ducts

An analytical approach based on the generalized integral transform technique is presented, for the solution of laminar forced convection within the thermal entry region of ducts with arbitrarily shaped cross-sections. The analysis is illustrated through consideration of a right triangular duct subjected to constant wall temperature boundary condition. Critical comparisons are made with results available in the literature, from direct numerical approaches. Numerical results for dimensionless average temperature and Nusselt numbers are presented for different apex angles.Nomenclature a,b sides of right triangular duct - A _c cross-sectional area of duct - c _p specific heat of fluid - D _h =4A _c /p hydraulic diameter, with P the wet perimeter - h(z) heat transfer coefficient at duct wall - k thermal conductivity - Pe=c _p D _h /k Peclet number - T(x, y, z) temperature distribution - T _o inlet temperature - T _w prescribed wall temperature - u(x, y); U(X, Y) dimensional and dimensionless velocity profile - average flow velocity - x; X dimensional and dimensionless normal coordinate (Fig. 1) - x ₁(y); X ₁(Y) dimensional and dimensionless position at irregular boundary (Fig. 1) - y; Y dimensional and dimensionless normal coordinate (Fig. 1) - z; Z dimensional and dimensionless axial coordinate Greek letters side of right triangular duct in X direction (dimensionless) - side of right triangular duct in Y direction (dimensionless) - density of fluid - (X, Y, Z) dimensionless temperature distribution - * apex angle of triangular duct (Fig. 1) - ** apex angle of triangular duct (Fig. 1)     

Thermal entrance region heat transfer for MHD laminar flow in parallel-plate channels with unequal wall temperatures

The problem of thermal entry heat transfer for Hartmann flow in parallel-plate channels with uniform but unequal wall temperatures considering viscous dissipation, Joule heating and axial conduction effects is approached by the eigenfunction expansion method. The series expansion coefficients for the nonorthogonal eigenfunctions are obtained by using a method for nonorthogonal series described by Kantorovich and Krylov [21]. Numerical results are obtained for the case with entrance condition parameter _o=1 and open circuit condition K=1. The parametric values of Ha=0, 2, 6, 10 and Br=0, –1 are considered for Hartmann and Brinkman numbers, respectively.

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Zusammenfassung Das Problem der Wärmeübertragung im thermischen Einlauf einer Hartmannströmung im ebenen Spalt mit einheitlichen, aber ungleichen Wandtemperaturen wurde unter Berücksichtigung viskoser Dissipation, Joulescher Heizung und axialer Wärmeleitung mit Hilfe einer Entwicklung nach Eigenfunktionen behandelt. Die Koeffizienten der Entwicklung für nichtorthogonale Eigenfunctionen wurde nach einer Methode für nichtorthogonale Reihen nach Kantorovicz und Krylow [21] berechnet. Numerische Ergebnisse werden für den Eintrittsparameter _o=1 und die Bedingung für den offenen Stromkreis K=1 erhalten. Die Parameterwerte Ha=0, 2, 6, 10 und Br=0, –1 werden für die jeweiligen Werte der Hartmann- und der Brinckman-Zahl betrachtet.

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Nomenclature a one-half of channel height - ¯B,B₀ magnetic field Induction vector and magnitude of applied magnetic field - Br Brinkman number, _f U_m ²/(k_c) - C_n,D_n coefficients in the series expansion of _e, see eq. 16 - c_p specific heat at constant pressure - ,E₀ electric field intensity vector and component - E_n,O_n even and odd eigenfunctions - Ha Hartmann number, (/_f)^1/2 B_o a - h₁,h₂ local heat transfer coefficients at lower and upper plates - ¯J,J_y electric current density vector and component - K external loading parameter, E_o/(B_o U_m) - k thermal conductivity - Nu₁, Nu₂ local Nusselt numbers, h₁,a/k and h₂a/k, respectively - P fluid pressure - Pe Peclet number, PrRe - Pr Prandtl number, C_{p f}/k - q₁,q₂ rates of heat transfer per unit area,^–k(T/Z)Z=–a'^–k(T/Z) Z=a respectively - Re Reynolds number, U_ma/u_f - T,T₀,T₁,T₂ fluid temperature, uniform entrance temperature, uniform but different lower and upper plate temperatures, respectively - T_b,T_m bulk temperature and (T₁+T₂)/2 - U,U_m,u axial, mean and dimensionless velocities, respectively - ¯V velocity vector - X,Z axial and transverse coordinates - x,z dimensionless coordinates - _n,_n even and odd eigenvalues - ,₀,_b dimensionless fluid, entrance and bulk temperatures, respectively - _c,_e,_f characteristic temperature difference (T₂-T_m), and dimensionless fluid temperatures, defined by eq. (10) - _e,_f magnetic permeability and viscosity of fluid - fluid density - electric conductivity - viscous dissipation function - (1-)/2     

Wave angle for oblique detonation waves

The flow field associated with a steady, planar, oblique detonation wave is discussed. A revision is provided for- diagrams, where is the wave angle and is the ramp angle. A new solution is proposed for weak underdriven detonation waves that does not violate the second law. A Taylor wave, encountered in unsteady detonation waves, is required. Uniqueness and hysteresis effects are also discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Temperature distribution in a thin rotating disk

The problem of heat conduction in a thin rotating disk with heat input at a fixed point is considered. The disk is cooled by forced convection from its lateral surfaces. By defining a complex temperature, the temperature throughout the disk is presented as a series of Bessel functions of complex argument. Results are given for a range of rotational speeds.Nomenclature R radial coordinate - angular coordinate - a radius of disk - b thickness of disk - T temperature - T ambient temperature - rotational speed of disk - q heat flux into disk - k thermal conductivity of disk - density of disk - c specific heat of disk - h coefficient of convective heat transfer - r dimensionless radial coordinate, R/a - T* characteristic temperature, q ₀ a/ k - t dimensionless temperature, (T–T )/T* - C ₁, C ₂ dimensionless parameters defined in (3)     

Simultaneous measurements of velocity and temperature fluctuations in thermal boundary layer in a drag-reducing surfactant solution flow

The mechanism of turbulent heat transfer in the thermal boundary layer developing in the channel flow of a drag-reducing surfactant solution was studied experimentally. A two-component laser Doppler velocimetry and a fine-wire thermocouple probe were used to measure the velocity and temperature fluctuations simultaneously. Two layers of thermal field were found: a high heat resistance layer with a high temperature gradient, and a layer with a small or even zero temperature gradient. The peak value of was larger for the flow with the drag-reducing additives than for the Newtonian flow, and the peak location was away from the wall. The profile of was depressed in a similar manner to the depression of the profile of in the flow of the surfactant solution, i.e., decorrelation between v and compared with decorrelation between u and v. The depression of the Reynolds shear stress resulted in drag reduction; similarly, it was conjectured that the heat transfer reduction is due to the decrease in the turbulent heat flux in the wall-normal direction for a flow with drag-reducing surfactant additives.List of symbols ensemble averaged value - (·)⁺ normalized by the inner wall variables - (·) root-mean-square value - C concentration of cetyltrimethyl ammonium chloride (CTAC) solution - c _p heat capacity - D hydraulic diameter - f friction factor - H channel height - h heat transfer coefficient - j _H Colburn factor - l length - Nu Nusselt number, h - Pr Prandtl number, c _p/ - q _w wall heated flux - Re Reynolds number, U _b/ - T temperature - T _b bulk temperature - T _i inlet temperature - T _w wall temperature - T friction temperature, q _w /c _p u - U local time-mean streamwise velocity - U ₁ velocity signals from BSA1 - U ₂ velocity signals from BSA2 - U _b bulk velocity - u streamwise velocity fluctuation - u1 velocity in abscissa direction in transformed coordinates - u friction velocity, - v wall-normal velocity fluctuation - v1 velocity in ordinate direction in transformed coordinates - var(·) variance - x streamwise direction - y wall-normal direction - z spanwise direction - _j junction diameter of fine-wire TC - _w wire diameter of fine-wire TC - angle of principal axis of joint probability function p(u,v) - _f heat conduction of fluid - _w heat conduction of wire of fine-wire TC - kinematic viscosity - local time-mean temperature difference, T _w –T - temperature fluctuation - standard deviation - density - _w wall shear stress     

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     

Thermal stability of composite superconducting tape under the effect of a two-dimensional dual-phase-lag heat conduction model

Thermal stability of composite superconducting tape subjected to a thermal disturbance is numerically investigated under the effect of a two-dimensional dual-phase-lag heat conduction model. It is found that the dual-phase-lag model predicts a wider stable region as compared to the predictions of the parabolic and the hyperbolic heat conduction models. The effects of different design, geometrical and operating conditions on superconducting tape thermal stability were also studied.a conductor width, (m) - A conductor cross sectional area of, (m²) - As conductor aspect ratio, (a/b) - b conductor thickness, (m) - Bi Biot number - B dimensionless disturbance Intensity - C heat capacity, (J m^–3 K^–1) - D disturbance energy density, (W m^–3) - f volume fraction of the stabilizer in the conductor - g(T) steady capacity of the Ohmic heat source, (W m^–3) - g_max Ohmic heat generation with the whole current in the stabilizer, (W m^–3) - G_max dimensionless maximum Joule heating - h convective heat transfer coefficient, (W m^–2 K^–1) - J current density, (A m^–2) - k thermal conductivity of conductor, (W m^–1 K^–1) - q conduction heat flux vector, (W m^–2) - Q dimensionless Joule heating - R relaxation times ratio (_T/2_q) - t rime, (s) - T temperature, (K) - T_c critical temperature, (K) - T_c1 current sharing temperature, (K) - T_i initial temperature, (K) - T_o ambient temperature, (K) - x, y co-ordinate defined in Fig. 1, (m) - thermal diffusivity (m² s^–1) - dimensionless time - _i dimensionless duration time - dimensionless y-variable - _o superconductor dimensionless thickness - dimensionless temperature - _c1 dimensionless current sharing temperature - 1 dimensionless maximum temperature - dimensionless disturbance energy - numerical tolerance - x width of conductor subjected to heat disturbances, (m) - y thickness of conductor subjected to heat disturbances, (m) - dimensionless x-variable - _o superconductor dimensionless width - stabilizer electrical resistivity, () - _q relaxation time of heat flux, (s) - _T relaxation time of temperature gradient, (s) - i initial - sc current sharing - max maximum - o ambient     

Hydrodynamical changes due to polymer migration in very dilute solutions

The slip hypothesis, based on thermodynamical arguments, has been extended to obtain the flow characteristics of polymer solutions flowing in a nonhomogeneous flow field. An asymptotic analysis, valid for both channel and falling film flows, is presented that predicts the flow enhancement due to polymer migration. Concentration-viscosity coupling is shown to be a critical factor in the hydrodynamic analysis. The analysis, which essentially provides an upper bound on flow enhancement, explicitly accounts for the influence of wall shear stress, initial polymer concentration etc. A comparison with the pertinent experimental data shows reasonable agreement. c concentration - c ₀ concentration in shear-free region - c _i initial concentration - d rate of deformation tensor - g acceleration due to gravity - g ₁ function defined in eq. [13] or [15] - g ₂ function defined in eq. [18] or [20] - H half-channel thickness or film thickness - K gas law constant - L length of the channel or film - q flow rate per unit width - q ^* normalized flow rate - T temperature - v velocity - V mean velocity - y transverse distance - y _c location of solvent layer - _w /µ _s - _w /c ₀ KT - /t convected derivative - dimensionless cenentration,c/c ₀ - _c dimensionless interface concentration - _w dimensionless wall concentration - relaxation time - µ _eff effective viscosity - µ _s solvent viscosity - dimensionless transverse distance,y/H - _c dimensionless interface location - density - stress tensor - _w wall shear stress - c _i KT/ _w - ns no slip NCL-Communication No. 3155     

Analysis of fully developed opposing mixed convection between inclined parallel plates

An exact solution is presented of fully developed, laminar flow between inclined parallel plates with a uniform wall heat flux boundary condition. The flow is downward and the heat flux is into the channel, so that natural convection opposes the forced flow. The solution depends on the two parametersP ₁=Gr sin/Re andP ₂=Gr cos/Re ² Pr. Four different flow reversal regimes are observed: 1) no reversal, 2) top reversal, 3) bottom reversal, and 4) top and bottom reversal. Velocity profiles, temperature profiles, wall friction, and Nusselt numbers are presented. Despite the simplicity of the problem which has been analyzed, it does display some features which have been observed in real mixed convection flows, such as flow reversal and nonmonotonic dependence on tilt angle.

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Berechnung der voll entwickelten entgegengesetzt gerichteten Misch-Konvektion zwischen geneigten parallelen Platten

Zusammenfassung Es wird eine exakte Lösung für voll entwickelte laminare Strömung zwischen geneigten parallelen Platten mit einheitlichem Wand-Wärmestrom als Randbedingung dargestellt. Die Strömung ist abwärts gerichtet und der Wärmestrom führt in den Kanal, so daß die freie Konvektion der erzwungenen entgegengesetzt gerichtet ist. Die Lösung hängt von den beiden Parametern P₁=Gr sin/Re und P₂=Gr cos/Re ² Pr ab. Vier verschiedene Bereiche der Strömungsumkehr wurden betrachtet: 1) keine Richtungsumkehr, 2) Umkehr an der Oberseite, 3) Umkehr an der Unterseite und 4) Umkehr an Ober- und Unterseite. Es wurden Geschwindigkeits- und Temperaturprofile, Wandreibung und Nusselt-Zahlen dargestellt. Trotz der Einfachheit des analysierten Problems werden einige Dinge dargestellt, welche in realer gemischter Konvektion untersucht wurden, so z.B. Strömungsumkehr und die nicht-monotone Abhängigkeit vom Schrägungswinkel.

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Nomenclature f friction factor=_w/(1/2_ou^–2) - g gravitational acceleration constant=9.8 m/s² - Gr Grashof number=qL⁴/kv² - k fluid thermal conductivity - L channel width - Nu Nusselt number=2qL/k(T _w –T _h ) - p fluid thermodynamic pressure - P nondimensional pressure=[p– _o g(x sin–y cos)]/Pr _o u ^– - Pr Prandtl number=v/ - P ₁ Gr sin/Re - P₂ Gr cos/Re ² Pr - q wall heat flux - Re Reynolds number=–u L/v - T fluid temperature - T _b fluid bulk temperature - T _o constant reference temperature - T _w fluid temperature - u axial velocity - average velocity - U nondimensional velocity=u/ - x axial coordinate - X nondimensional axial coordinate=x/–uL ² - y transverse coordinate - Y nondimensional transverse coordinate=y/L - fluid thermal diffusivity - fluid thermal expansion coefficient - tilt angle, measured counterclockwise from horizontal in Fig. 1 - v fluid kinematic viscosity - ₀ fluid density evaluated atT ₀ - _w wall shear stress - nondimensional temperature=(T–T _o )/(q L/k) - _b nondimensional bulk temperature=f U dY - _w nondimensional wall temperature=(T _w –T ₀)/(qL/k)     

Visualization study of evaporation of single n-pentane drops in water

A laser shadowgraph system was constructed to enable successive filming of a drop or a bubble rising or falling in an immiscible liquid confined within a vertical column. The assembly was applied to a study of the evaporation of n-pentane drops in a stagnant medium of water. The liquid/vapor two-phase bubble evolving from each pentane drop was observed together with its wake, the morphology and the dynamics of which are our primary concern in considering the mechanism of the medium-to-bubble heat transfer.List of symbols a minor axis of ellipsoidal two-phase bubble - b major axis of ellipsoidal two-phase bubble - D ₀ diameter of saturated-liquid drop set to vaporize - Re Reynolds number based on instantaneous, volume-equivalent spherical diameter and rise velocity of two-phase bubble and kinematic viscosity of the continuous phase - t time lapse after the start of evaporation - T ^* excess of undisturbed continuous-phase temperature above the temperature at which the sum of the saturated vapor pressures of the dispersed- and the continuous-phase fluids is equal to the pressure at the position where evaporation starts - opening angle of wake-covered region on bubble surface - _w zenith angle at flow-separation ring on bubble surface - mass fraction of vapor in two-phase bubble     

Linearized analysis of magnetohydrodynamic entrance flow in a vertical channel with combined thermal convection

A linearized analysis is presented for the magnetohydrodynamic entrance flow with combined forced and free convection in a vertical, constant wall temperature parallel-plate channel. Numerical results are obtained for slug velocity profile at the entrance and for various Hartmann and Grashof Numbers. The results agree well with the finite difference numerical solutions obtained elsewhere. They demonstrate that the velocity development and pressure gradient in the channel entrance region are greatly influenced by the Hartmann Number and the Grashof Number. Increasing Hartmann Number decreases velocity entrance length while increasing Grashof Number increases it. Thermal development is also found to be dependent on the above mentioned parameters, but to a relatively minor extent.Nomenclature A _m constant defined by equation (23) - B ₀ applied magnetic field - C _n constant defined by equation (13) - E ₀ constant electric field - e nondimensional electric field parameter, E ₀/U _mB₀ - Gr Grashof Number, gL ³(T _w–T ₀)/ ² - L half-width of the channel - M Hartmann Number, B ₀ L(/)^1/2 - Nu Nusselt Number, (/y) _y=1/( _w– _m) - P pressure - Pr Prandtl Number, / - p nondimensional pressure parameter, (P–P ₀+ ₀ gX)/P ₀ U _m ² - Re Reynolds Number, U _m L/ - T temperature - T ₀ inlet temperature - T _w wall temperature - U velocity, X direction - U _m average velocity, (1/L) ₀ ^L UdY - u nondimensional form of U, U/U _m - u ₀(y) nondimensional inlet velocity - V velocity, Y direction - v nondimensional form of V, VL/ - X coordinate, axial direction - x nondimensional form of X, vX/L ² U _m - Y coordinate perpendicular to the channel - y nondimensional form of Y, Y/L - thermal diffusivity - _m eigenvalue defined by equation (25) - thermal expansion coefficient - _m eigenvalue defined by equation (24) - stretching factor, weighting function - nondimensional form of T, (T–T ₀)/(T _w–T ₀) - _m mean nondimensional temperature, ₀ ¹ udy - kinematic viscosity - magnetic permeability - mass density - electrical conductivity     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

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Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

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Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

Effects of viscous dissipation on heat transfer in an oscillating flow past a flat plate

Theoretical investigation has been carried out of laminar thermal boundary layer response to harmonic oscillations in velocity associated with a progressive wave imposed on a steady free stream velocity and convected in the free stream direction. Series solutions are derived both to velocity and temperature field and the resulting equations are solved numerically. The functions affecting the temperature field are shown graphically for different values of Prandtl number. It is observed that there is more reduction in the rate of heat transfer for P _r<1 and a rise in the rate of heat transfer for P _r>1 due to the presence of oscillatory free-stream.Nomenclature u, v velocity components in the x and y direction - x, y Cartesian coordinate axes - t time - U, U ₀ instantaneous value of and mean free stream velocity - density of fluid - kinematic viscosity - T, T _w, T temperature of the fluid, wall and free stream fluid - c _p specific heat at constant pressure - thermal diffusivity - amplitude of free stream velocity - frequency - _p non-dimensional temperature (T–T /T _w–T ) - P _r Prandtl number (c _p/K) - E _c Eckert number (U ₀ ² /c _p(T _w–T )) - a parameter ( ) - ₀ boundary layer thickness of the oscillation of a harmonic oscillation of frequency ( ) - ordinary boundary layer thickness ( ) - time-averaged, time-independent external velocity - A, B, C, D, E, K, L, M, N, P functions used in expansion for u and - Nu Nusselt number (hx/k) - T _w–% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8V4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcadaGcaa% qaaiaadAhacaWG4bGaai4laiqadwfagaqeaaWcbeaakiaacMcaaaa!3CA6!\[(\sqrt {vx/\bar U} )\] - k thermal conductivity     

Anlaufkorrektur und Elastizität bei der Kapillarströmung von Lösungen

Zusammenfassung Aus der Anlaufkorrektur kann man nach einer Rechnung vonFromm, die auf dem Maxwellschen Modell basiert, eine Relaxationszeit und eine korrigierte Viskosität _c ermitteln. Der Quotient _c/ stellt einen Schermodul dar. Diese Größe wird für Lösungen von Cellulosetrinitrat in Butylacetat, Polyvinylacetat in Dioxan, Polystyrol in Toluol, Polyacrylamid in Wasser, und Viskose, in Abhängigkeit von der Konzentrationc und dem SchergefälleD ermittelt. Es zeigt sich, daß _c/ etwa im Wendepunkt der Fließkurven eine Art Plateau oder ein flaches Maximum zeigt und in diesem Plateaubereich eine lineare Abhängigkeit von der Konzentration. Die absolute Größe von _c/ ist jedoch um Größenordnungen geringer, als sie nach der Formel vonRouse bzw.Bueche für die erste Relaxationszeit eines Verhängungsnetzwerkes zu erwarten wäre. Das wird so gedeutet, daß bei dem hohen Schergefälle, das bei den Messungen herrschte (D etwa 10⁴ sec^–1), ein Teil der Verhängungen zerstört ist, wodurch die Relaxationszeit vergrößert und der Schermodul verkleinert wird.

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Summary From the end-correction, according to a calculation byFromm based upon theMaxwell-model, a relaxation time and a corrected viscosity _c can be obtained. The quotient _c/ represents a shear modulus. Its value is determined for solutions of cellulosetrinitrate in butylacetate, polyvinylacetate in dioxane, polystyrene in toluene, polyacryloamide in water, and viscose, in dependence of concentrationc and shear rateD. It is found, that _c/ shows a plateau or a flat maximum at the inflection point of the flow curves. In this range, a linear dependence on concentration is found too. The absolute value of _c/, however, is smaller by orders of magnitude than that to be expected for the first relaxation time of an entanglement network according to the formulas byRouse resp.Bueche. This is explained by a partial disruption of entanglements in the high shear rate prevailing at the experiments (D about 10⁴ sec^–1), which effects an increase of the relaxation time and a decrease of the shear modulus.

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Vorgetragen auf der Jahrestagung der Deutschen Rheologen in Bad Ems vom 18.–19. Mai 1967.     

An experimental study of the behavior of transient isolated convection in a rigidly rotating fluid

The temperature field of starting thermal plumes were measured in a rotating annulus with various rotation rates and buoyancies. The experiments revealed many details of the internal structure of these convective phenomena and also significant horizontal displacements from their source. Measurements show an increase in the maximum temperature observed in the thermal caps with increasing rotation and a more rapid cooling of the buoyancy source.List of symbols D angle relating inward centripetal acceleration to buoyant acceleration, defined by tan D = R/g - g gravitational acceleration - P total pressure of ambient fluid - R radial coordinate measured from rotation axis - R ₀ distance from rotation axis to buoyancy source - u velocity of fluid parcel along the radial direction - velocity of fluid parcel along the azimuthal direction - w velocity of fluid parcel along the axial direction - z axial coordinate, measured upward from the plane containing the buoyancy source - density of a buoyant parcel of fluid - ₀ density of the ambient fluid - azimuthal angle measured from the radial line passing through the buoyancy source - rotation rate of the R––z coordinate system in radians/second