The freeze-shut of a convectively cooled parallel plate channel subjected to laminar internal liquid flow

The paper presents an approximative solution for the time dependent development of the ice layers at the cooled walls inside a parallel plate channel. The upper and the lower wall of the channel are cooled by an uniform external convection. By assuming a constant pressure drop across the channel, the freeze-shut of the planar channel could be calculated approximately. It was found out that the origin of the freezing fronts moves upstream during the ice layer growth. Furthermore a simple criterion is presented to predict whether a given system will lead to blockade.     

Transient temperature measurements in a convectively cooled droplet

 The transient cooling of an evaporating water droplet, suspended in a jet of dry air, was experimentally investigated in this study using thermochromic liquid crystal thermography. Microencapsulated beads of thermochromic liquid crystals, suspended in the water droplets, enabled the visualization of the transient temperature fields within the droplets. Digital movies of the convectively cooled droplets reveal spatial and temporal temperature gradients resolved down to length scales of ∼100 μm and time scales of ∼0.03 s. The transient temperature measurements were analyzed to yield total droplet convective heat transfer rates. Droplet heat transfer rates determined from a heat balance on the droplets compare favorably to previously published measurements. Received: 11 June 1997/Accepted: 26 March 1998     

Inward solidification of a superheated liquid in a cooled horizontal tube

Inward solidification has been studied experimentally and analytically under conditions where the liquid phase is above the fusion temperature (i.e., superheated). The liquid was housed in a horizontal circular tube in which the surface was maintained at a uniform, time-invariant temperature during test runs. Three phase change materials (n-heptadecane,n-octadecane, and water) were used in the tests. Both analysis and experiments have established that for inward solidification, natural convection in a superheated liquid is not important in controlling the solidliquid interface motion for Stefan numbers less than unity. The interface velocity is determined primarily by the thermal resistance across the solid layer. Good agreement has been obtained between experimentally measured and analytically predicted solid-liquid interface positions when the density differences between the phases were accounted for.     

Heat transfer and solidification of a laminar liquid flow in a cooled parallel plate channel: The stationary case

A simple numerical model is presented to predict the steady-state ice layers on the cooled walls inside a parallel plate channel for arbitrary entrance velocity profiles. The effect of two different entrance velocity distributions (a parabolic velocity distribution and a slug flow) on the shape of the ice-layers are examined. The quality of an approximative solution given in literature was checked by comparing with the numerical results. For the case of a fully developed parabolic velocity distribution at the entrance of the cooled channel the results are compared with experimental findings of Kikuchi [8]. A generally good agreement was found.Es wurde ein einfaches numerisches Modell entwickelt, das es ermöglicht, die stationären Erstarrungsfronten an den Kanalwänden für beliebige Verteilungen des Eintrittsgeschwindigkeitsprofils zu berechnen. Als Beispiele wurden ein voll ausgebildetes Parabelprofil und ein Pfropfenprofil am Eintritt in die Kühlstrecke untersucht. Mit Hilfe der numerischen Lösung konnte die Güte einer aus der Literatur bekannten Näherungslösung zur Berechnung der Erstarrungsfronten überprüft werden. Für den Fall des Parabelprofils am Kanaleintritt wurde die Rechnung mit Meßwerten von Kikuchi [8] verglichen. Es zeigte sich eine gute Übereinstimmung zwischen Theorie und Experiment.     

Parametric solutions of the eigenvalue equation for a convectively cooled slab

The convective cooling of a slab by an ambient fluid under the most general linear and homogeneous boundary conditions is considered. For the roots of the corresponding transcendental eigenvalue equation an explicit formula is written down in a parametric form. The practical consequences of this representation, among them certain “singular solutions” which cannot be obtained by a direct numerical treatment of the original eigenvalue equation, are discussed in detail.     

Dynamics of particles floating on liquid flowing in a semicircular open channel

The dynamic characteristics of surface-floating particles in liquids flowing in a two-dimensional, semicircular open channel is studied experimentally. For high visibility in the experiments, relatively large particles are employed whose particle-liquid density ratio is either equal to or less than unity. Particles of different size and geometry are tested in a water-glycerin mixture. A video camera traces the pathline of each particle from which the velocity and direction of particle motion are evaluated. Liquid velocity distribution is determined by hot-film anemometry. A modified dynamics (Basset-Boussinesq-Oseen) equation is derived and numerically solved by means of a finite-difference technique to determine fluid velocity. A new dimensionless parameter is disclosed which is pertinent to both particle geometry and fluid flow conditions. It correlates particle trajectory and velocity, trajectory dispersion and fluid-particle velocity ratio.Visiting Scholar on leave from Department of Mechanical Engineering, Fukuyama University, Fukujama, Japan     

Heat transfer in a radiating fluid with slug flow in a parallel-plate channel

Summary As a step towards a better understanding of combined conduction, convection, and radiation, fully developed heat transfer in slug flow in a flat duct between two parallel plates is considered. The flat duct consists of two diffuse, nonblack, isothermal surfaces. The gray radiating fluid between them is capable of emitting and absorbing thermal radiation. The problem is formulated in terms of a nonlinear integrodifferential equation, and the solution is obtained by an approximate method. The differences between heating and cooling the fluid are examined. The effects of the optical thickness of the fluid, the ratio which determines the relative role of energy transport by conduction to that by radiation, and the emissivity of the duct surfaces on the temperature distribution and the heat transfer characteristics are investigated. An approximate method for calculating the radiant heat flux at the wall is presented, and the accuracy of the approximation is tested.Work done under the auspices of the U.S. Atomic Energy Commission.     

A theoretical and experimental investigation of smooth- and wavy ice layers in laminar and turbulent flow inside an asymmetrically cooled parallel-plate channel

The present paper shows the adaption of the numerical model originally developed by Weigand and Beer [14] for calculating steady-state ice layers inside an asymmetrically cooled parallel-plate channel. The investigation shows the characteristics in ice formation behaviour due to asymmetrically cooled walls. Further, a simple analytical model is presented for calculating smooth ice layers in turbulent flow. The study is supported by own measurements of the freezing fronts inside an asymmetrically cooled channel. A comparison between theoretical calculations and measurements shows generally good agreement.Die vorliegende Arbeit beschreibt die Anwendung des von Weigand und Beer [14] entwickelten, numerischen Modells zur Vorhersage von Eisschichten in einem ebenen, asymmetrisch gekühlten Kanal. Die Studie befaßt sich mit den Unterschieden in der Eisschichtbildung aufgrund der asymmetrisch gekühlten Kanalwände. Weiterhin wird ein einfaches Verfahren angegeben, mit dem sich die Gestalt von glatten Eisschichten bei turbulenter Strömung und asymmetrischer Kühlung sehr einfach berechnen läßt. Die analytisch und numerisch gewonnenen Resultate werden anschließend mit eigenen Messungen von Eisschichten verglichen, wobei eine im allgemeinen gute Übereinstimmung zwischen Theorie und Experiment zu beobachten ist.     

Technical stability in a straight pipeline containing a flowing liquid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 148–155, January–February, 1990.     

Combined free and forced convection flow in a cooled vertical duct with internal solidification

The effect of mixed convection flow on the shape of the frozen crust in a cooled vertical channel was investigated numerically. For the prediction of the ice-layer thickness a simple numerical model which is based on the boundary layer equations was used. It can be seen that in case of assisting mixed convection flow the heat transfer at the solid crust increases because of inreasing velocity near the solid-liquid interface. On the other hand this increase of the velocity near the solid-liquid interface can lead to flow separation in the core region of the channel because of continuity of mass. By comparing the numerically obtained results for aiding mixed flow with measurements of Campbell and Incropera [10] good agreement can be observed. In case of opposing mixed flow it can be shown that flow separation might occur near the solid-liquid interface. This can result in a wave-like structure of the ice-layer.     

Asymptotic Nusselt numbers for Newtonian flow through a parallel-plate channel with recycle

General expressions for evaluating the asymptotic Nusselt number for a Newtonian flow through a parallel-plate channel with recycle at the ends have been derived. Numerical results with the ratio of thicknesses as a parameter for various recycle ratios are obtained. A regression analysis shows that the results can be expressed by log Nur^0.83=0.3589 (log)² -0.2925 (log) + 0.3348 forR 3, 0.1 0.9; logNu=0.5982(log)² +0.3755 × 10^–2 (log) +0.8342 forR 10^–2, 0.1 0.9.

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Asymptotische Nusselt-Zahlen für die Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung

Zusammenfassung In dieser Untersuchung wurden allgemeine Ausdrücke hergeleitet um die asymptotische Nusselt-Zahl für eine Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung an den Enden berechnen zu können. Es wurden numerische Ergebnisse mit den Dicken-Verhältnissen, als Parameter für verschiedene Rückführungs-verhältnisse, erhalten. Eine Regressionsanalyse zeigt, daß die Ergebnisse wie folgt ausgedrückt werden können: log Nur^0,83=0,3589 (log)² -0,2925 (log) + 0,3348 fürR 3, 0,1 0,9; logNu=0,5982(log)² +0,3755 × 10^–2 (log) + 0,8342 fürR 10^–2, 0,1 0,9.

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Nomenclature A₁ shooting value,d(0)/d - A₂ shooting value,d(1)/d - B channel width - Gz Graetz number, U_bW²/L - h _m logarithmic average convective heat transfer coefficient - h _x average local convective heat transfer coefficient - k thermal conductivity - L channel length - Nu average local Nusselt number, 2 h_xW/k - Nu _m logarithmic average Nusselt number, 2h_mW/k - R recycle ratio, reverse volume flow rate divided by input volume flow rate - T temperature of fluid - T _m bulk temperature, Eq. (8) - T ₀ temperature of feed stream - T _s wall temperature - U velocity distribution - U _b reference velocity,V/BW - V input volume flow rate - v dimensionless velocity distribution, U/U_b - W channel thickness - x longitudinal coordinate - y transversal coordinate - Z₁-z₆ functions defined in Eq. (A1) - thermal diffusivity - least squares error, Eq. (A7) - weight, Eqs. (A8), (A9) - dimensionless coordinate,y/W - dimensionless coordinate,x/GzL - function, Eq. (7)     

Thermosolutal mixed convection and flow-reversal in an inclined parallel-plate channel

The steady-state laminar mixed convection of a binary gas mixture in a parallel-plate channel is investigated. The channel walls are subjected to different combinations of first-type thermal and solutal boundary conditions and different wall inclinations have been considered. A second-order accurate control-volume based numerical scheme is used for the resolution. In parallel with the numerical investigation, the governing conservation equations are also simplified for fully developed conditions and are shown to be controlled by a single parameter. An exact analytical solution is obtained for the main flow variables and transfer rates and serves as a validation tool for the numerical model. In addition, it establishes a criterion based on the two Grashof numbers, the Reynolds number and the channel inclination for the existence of a reversed flow.     

Simulation of metal matrix composite solidification in the presence of cooled fibers

Metal matrix composite (MMC) has been well known for its superior material properties compared with traditional composite. A new method is introduced to improve the properties of MMC in the sense that the ends of the reinforcement phase of the composite are allowed to extend out of the mold and cooled by a heat sink in order to promote the rate of heat transfer through the fibers and promote the formation of primary alpha phase around the reinforcement. This paper presents the experimental results obtained from the foundry in the University of Wisconsin-Milwaukee and some numerical simulation results of the solidification process in the cast mold.     

Thermophoretic deposition of particles in fully developed mixed convection flow in a parallel-plate vertical channel

A theoretical analysis is made for thermophoretic transport of small particles through a fully developed laminar, mixed convection flow in a parallel vertical channel. The governing gas-particle ordinary differential equations are expressed in non-dimensional form and are solved numerically for some values of the governing parameters so as to investigate extensively their distinct influence on the flow pattern. These equations are solved also analytically in the special case when the thermophoretic effect is absent and the obtained analytical solution can be regarded as a verification of the numerical results, simultaneously. The parameter zone for the occurrence of reversed flow is presented. It is found that the effect of thermophoretic can be quite significant in appropriate situations.     

Stability of a viscous liquid film flowing down a periodic surface

The paper is devoted to a theoretical analysis of linear stability of the viscous liquid film flowing down a wavy surface. The study is based on the Navier–Stokes equations in their full statement. The developed numerical algorithm allows us to obtain pioneer results in the stability of the film flow down a corrugated surface without asymptotic approximations in a wide range over Reynolds and Kapitsa’s numbers. It is shown that in the case of moderate Reynolds numbers there is a region of the corrugation parameters (amplitude and period) where all disturbances decay in time and the wall corrugation demonstrates a stabilizing effect. At the same time, there exist corrugation parameters at which the steady-state solution is unstable with respect to perturbations of the same period as the period of corrugation. In this case the waveless solution cannot be observed in reality and the wall corrugation demonstrates a destabilizing effect.     

Lifetime of a narrow channel in a liquid

Lifetime determination is considered for a narrow channel in a liquid collapsing in response to gravitational and capillary forces. Working formulas are derived to relate the lifetime to the properties of the liquid and the channel parameters. The calculations are compared with experiments on the effects of a focused high-power electron beam acting on a liquid and solid.Translated from Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 121–129, January 1974.