Convection regime flow in a vertical slot: continuum of solutions from capped to open ends

In a recent publication Bühler (Heat Mass Transfer 39:631–638, 2003) reported new results for conduction regime flow between vertical differentially-heated walls that provide a continuum of solutions between capped and open ends. In this paper we extend Bühler’s work to realize a continuum of solutions of convection regime flow using empirical results for the vertical temperature gradient that develops in tall aspect ratio geometries. The mass flux is determined analytically for this three-parameter family of solutions. Identical viscous and thermal boundary layers exist at the opposing walls when the cavity is capped. However, as the flow evolves to one with open ends, there is an intensification (attenuation) of the boundary layers near the hot (cold) walls. In the limit corresponding to an open-ended cavity, the boundary layer at the cold wall vanishes altogether.     

Analytical solution for transient laminar fully developed free convection in vertical channels

Using Green's function method, analytical solutions for transient fully developed natural convection in open-ended vertical circular and two-parallel-plate channels are presented. Different fundamental boundary conditions for these two configurations have been investigated and the corresponding fundamental solutions are obtained. These fundamental solutions may be used to obtain solutions satisfying more general thermal boundary conditions. In terms of the obtained unsteady temperature and velocity profiles, the transient volumetric flow rate, mixing cup emperature and local nusselt number are estimated.Zusammenfassung Für oben und unten offene vertikale Kanäle mit Kreisquerschnitt bzw. als Parallelplattenanordnung werden unter Verwendung der Methode der Greenschen Funktionen analytische Lösungen für die nichtstationäre, vollausgebildete, natürliche Konvektion gefunden und zwar unter Zugrundelegung verschiedener Fundamental-Randbedingungen bezüglich beider Konfigurationen. Die so ermittelten Fundamentallösungen können zur Gewinnung von Lösungen für allgemeine Randbedingungen dienen. Der zeitlich veränderliche Volumenstrom, die Mischtemperatur und die Nusselt-Zahl werden mit Bezug auf die erhaltenen nichtstationären Profile für Temperatur und Geschwindigkeit näher analysiert.

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Analytische Lösung für die nichtstationäre vollausgebildete laminare freie Konvektion invertikalen Kanälen

Nomenclature a local heat transfer coefficient based on the area of the heat transfer surface,q/(T _w –T ₀)=±(T/y)_w/(T_w–T₀), minus and plus signs apply respectively for heating and cooling in case of parallel-plate channel and vice versa in case of a tube - average heat transfer coefficient over the channel - c _p specific heat of fluid at constant pressure - f volumetric flow rate, for circular channels and or two-parallel-plate channels - F dimensionless volumetric flow rate,f/(2lvGr^*) for circular channels forfw/(lvGr ^*) for two-parallel-plate channels - g gravitational body force per unit mass (acceleration) - G Green's function - Gr Grashof number,±g(T _w–T₀)w³/v² in case of an isothermal boundary of±gqw ⁴/2kv² in case of a uniform heat flux (UHF) on the heat transfer boundary, the plus and minus signs apply to upward (heating) and downward (cooling) flows, respectively. ThusGr is a positive number in both cases. - Gr ^* modified Grashof number,wGr/l - h heat gained or lost by fluid from the entrance up to a particular elevation in the channel, ₀ fc _p(T _m–T ₀) for all cases - J ₀ Bessel function of zero order - k thermal conductivity of fluid - l height of channel - L dimensionless height of channel,1/Gr ^* - Nu local Nusselt number,|a| w/k - average Nusselt number, - p pressure of fluid inside the channel at any cross-section - p pressure defect at any point,p–p _s - p ₀ pressure of fluid at the channel entrance - p _s hydrostatic pressure, ₀ gz where the minus and plus signs are for upward (heating) and downward (cooling) flows, respectively - p dimensionless pressure defect at any point(pw ⁴)/(₀ l ²² Gr ²) - Pr Prandtl number,c _p/k - q heat flux at the heat transfer surface,q=±k(T/y)_w where the minus and plus signs are, respectively, for cooling and heating in case of circular pipe and vice versa in case of a parallel-plate channel - Ra Rayleigh number,GrPr - Ra ^* modified Rayleigh number,Gr ^*Pr - t time - T fluid temperature at any point - T _m mixing-cup (mixed-mean) temperature over any cross section, for circular channels, and for two-parallelplate channels - T ₀ initial and channel-inlet fluid temperature - T _w temperature of the heat-transfer wall - u axial velocity component at any point - U dimensionless axial velocity,uw ²/(lvGr^*) - w radius of circular tube or width (between plates) of parallel-plate channel - y radial or transverse coordinate - y dimensionless radial or transverse coordinate,y/w - z axial coordinate - Z dimensional axial coordinate,z/(lGr ^*) Greek symbols constant appears in Eq. (8) - parameter appears in Eq. (9) which equals the integration of with respect to or volumetric coefficient of thermal expansion - _n eigenvalues - parameter appears in Eq. (7) - _n eigenvalues - parameter appears in Eq. (12) - _n eigenvalues - parameter appears in Eq. (9) - dimensionless temperature,(T–T ₀)/(T_w–T₀) in case of an isothermal heat transfer boundary and(T–T ₀)/(qw/2k) for UHF boundary - _m dimensionless mixing cup temperature,(T _m–T₀)/(T_w–T₀) in case of an isothermal heat transfer boundary and(T _m–T₀)/(qw/2k) for UHF boundary - _w dimensionless temperature of the heat-transfer wall, equals unity in case of an isothermal heat transfer boundary and(T _w–T₀)/(qw/2k) for a UHF boundary - _n eigenvalues - dynamic viscosity of fluid - kinematic viscosity of fluid, /₀ - fluid density at temperatureT,₀[1–(T–T ₀)] - ₀ fluid density atT ₀ - demensionless time,tk/(cw²)     

Solution of fully developed free convection of a micropolar fluid in a vertical channel by homotopy analysis method

In this paper, we reconsider the problem of fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel with asymmetric wall temperatures and concentrations. The resulting boundary‐value problem is solved analytically by the homotopy analysis method. The accuracy of the present solution is found to be in excellent agreement with the solutions of Cheng (Int. Commun. Heat Mass Transfer 2006; 33 :627–635). Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Numerical study of natural convection flow in a vertical slot

The stability of convective motion, generated by a lateral temperature difference across a vertical slot, is studied numerically over a range ofGr=5000 to 1.5 × 10⁵,Pr=0.01 to 10, andA=8,16 and 20. Various cellular flow structures and temperature patterns are illustrated. Several branches of solutions characterized by different numbers of the cells in the flow patterns as well as by both steady and unsteady multicellular patterns are found for low-Prandtl-number fluid in the vertical slot. Meanwhile, the behaviors of the temperature variation and heat transfer are also discussed. The project supported by the National Natural Science Foundation of China (59776011) and by the Returnee from Abroad Funding of Academia Sinica.     

Viscous dissipation effects on fully developed combined free and forced non-Newtonian convection in a vertical tube

An investigation which includes the simultaneous effects of viscous dissipation and combined free and forced laminar non-Newtonian convection is presented. The problem under consideration is that of fully developed upflow in a vertical, circular tube which is heated with a constant wall heat flux. All properties are assumed to be constants in the analysis except for a temperature dependent density in the body force term which generates the free convection effects. The coupled continuity, momentum, and energy equations are solved using a finite difference technique. Numerical solutions are presented as a function of the parameters of the problem-flow behavior index n, Grashof number over Reynolds number ratio Gr/Re, and the Eckert number-Prandtl number product E Pr. The results show that heating due to viscous dissipation distorts the velocity profile, increases the friction factor, and decreases the Nusselt number.     

Instability of fully developed mixed convection with viscous dissipation in a vertical porous channel

Flow driven by an externally imposed pressure gradient in a vertical porous channel is analysed. The combined effects of viscous dissipation and thermal buoyancy are taken into account. These effects yield a basic mixed convection regime given by dual flow branches. Duality of flow emerges for a given vertical pressure gradient. In the case of downward pressure gradient, i.e. upward mean flow, dual solutions coincide when the intensity of the downward pressure gradient attains a maximum. Above this maximum no stationary and parallel flow solution exists. A nonlinear stability analysis of the dual solution branches is carried out limited to parallel flow perturbations. This analysis is sufficient to prove that one of the dual solution branches is unstable. The evolution in time of a solution in the unstable branch is also studied by a direct numerical solution of the governing equation.     

Fully developed magneto convection flow in a vertical rectangular duct

An analysis is performed to study the MHD free convection flow in a vertical rectangular duct for laminar and fully developed regime taking into consideration the effects of Ohmic heating and viscous dissipation. Numerical solutions are found using finite difference method of second-order accuracy. The effects of various physical parameters such as Hartmann number, aspect ratio, buoyancy parameter and circuit parameter are presented graphically. It is found that as Hartmann number, buoyancy parameter and aspect ratio increase, the upward and downward flow rates are increased for open circuit but decrease for short circuit.     

Laminar thermal convection in a vertical slot

The validity of the conduction regime approximation is studied in a vertical slot of height to width aspect ratio equal to 28. Velocity profiles are determined by laser-Doppler velocimetry (LDV) in water and hexane near room temperature, but also in water at mean temperatures close to its density maximum. Finally a glycerol-water mixture is experimentally studied in order to check the theoretical prediction taking into account the temperature dependence of the viscosity (non-Boussinesq effects). The measured velocity profiles and the computed ones, both analytically for an infinite layer or by 2D numerical simulations, are in excellent agreement for all Grashof numbers. When the density is a quadratic function of temperature, a criterion is derived for the existence of a bicellular convective regime. The validity of this criterion is also experimentally verified. Finally, for a highly viscous fluid, the conduction regime approximation is found to be valid for an aspect ratio greater than 4.     

Thermophoretic deposition of particles in fully developed mixed convection flow in a parallel-plate vertical channel: the full analytical solution

In a recent paper (Grosan et al. in Heat Mass Transf 45:503–509, 2009) a mostly numerical approach to the title problem has been reported. In the present paper the full analytical solution is given. Several new features emerging from this approach are discussed in detail.     

Higher approximations to the free convection flow from a heated vertical flat plate

This paper is concerned with the problem of obtaining higher approximations for the free convection from a heated vertical flat plate to that represented by the well known solution of Schmidt and Beckmann. For large Grashof number, the perturbation problem is a singular one and the method of matched asymptotic expansions is used to construct inner and outer expansions for the velocity and temperature distributions. The small perturbation parameterε is the inverse of the fourth root of the Grashof number and the expansions are shown to involve only integral powers ofε. The first three terms in the expansion are calculated and numerical results are presented for the velocity, temperature, skin friction and heat transfer. The agreement with experiment is found to be excellent, and the theory fully explains the discrepancies which exist between boundary layer theory and experiment.     

Finite difference analysis of laminar free convection flow past a non isothermal vertical cone

A finite-difference analysis for the transient free convection flow of an incompressible viscous fluid past a vertical cone with variable wall surface temperature T _w^′ (x) = T _∞^′ + a x ⁿ varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters such as Prandtl number and n (exponent in power law variation in surface temperature). The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.     

Stability of laminar flow of liquid in vertical layers with free convection

We give the results of experimental and theoretical investigations of the stability of a laminar flow of liquid in a vertical layer. The experimental investigations qualitatively confirm the theoretical solutions and indicate the existence of various kinds of instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 170–174, September–October, 1971.     

Steady two-phase flow in a vertical tube subject to combined free and forced convection

Using the two-velocity, two-temperature model of a continuous medium, the viscousgravitational flow of a mixture of incompressible liquid and solid particles in a vertical round tube is considered. The free-convection equations are written down on the basis of the general equation of motion and the energy equation of a two-phase medium [1, 2]. Using a finite Hankel integral transformation, a solution is constructed for the case of a linear wall-temperature distribution along the tube. The results of some practical calculations of the velocity and temperature fields over the cross section of the tube are presented, together with the dimensionless heat-transfer coefficient expressed as a function of the Rayleigh number and phase concentration. Here it is assumed that the dynamic and thermal-interaction coefficients between the phases correspond to the Stokes mode of flow for each particle, as a result of which the velocity and thermal phase lag is very small [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 132–136, July–August, 1975.     

Transient free convection flow of a viscoelastic fluid over vertical surface

In this paper, the viscoelsatic boundary layer flow and the heat transfer near a vertical isothermal impermeable surface and in a quiescent fluid are examined. The gov-erning equations are formulated and solved numerically using MackCormak’s technique. The results show excellent agreement with previously published results by a compari-sion. Representative results for the velocity and temperature profiles, boundary layer thicknesses, Nusselt numbers, and local skin friction coefficients are shown graphically for different values of viscoelsatic parameters. In general, it is found that the velocities increase inside the hydrodynamic boundary layers and the temperatures decrease inside the thermal boundary layers for the viscoelsatic fluid as compared with the Newtonian fluid due to favorable tensile stresses. Consequently, the coefficients of friction and heat transfer enhance for higher viscoelsatic parameters.     

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.     

Unsteady mixed convection boundary layer flow over a vertical cone with non-uniform slot suction (injection)

The non-similar solution of an unsteady mixed convection laminar boundary layer flow over a vertical cone in the presence of non-uniform surface mass transfer through slot has been obtained while the axis of cone is inline with the flow. The unsteadiness is caused by the time dependent free stream velocity. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting coupled non-linear partial differential equations have been solved numerically by the combination of quasi-linearization technique and an implicit finite difference scheme. Numerical computations are performed for different values of the parameters to display the velocity and temperature profiles graphically. Both accelerating and decelerating free stream velocities are considered. Numerical results are reported to display the effects of non-uniform single and double slot suction (injection) on skin friction and heat transfer coefficients at the wall. Further, the effects of Prandtl number, buoyancy and mass transfer (suction or injection) parameters at different stream-wise locations for various times on velocity and temperature profiles, and on skin friction and heat transfer coefficients are also presented in this paper.