Numerical study of the natural convective cooling of a vertical plate

We study numerically in this paper the natural convective cooling of a vertical plate. The full transient heat conduction equation for the plate, coupled with the natural convection boundary layer equations are solved numerically for a wide range of the parametric space. Assuming a large Rayleigh number for the natural convection flow, the balance equations are reduced to a system of three differential equations with three parameters: the Prandtl number of the fluid, Pr, a non-dimensional plate thermal conductivity α and the aspect ratio of the plate ?. The nondimensional cooling time depends mainly on α/?², obtaining a minimum of this time for values of 1?α??².     

Thermophoresis particle deposition effects in a free convective doubly stratified medium over a vertical plate

An analysis is performed to study the thermophoresis effects in a transient free convective flow of a viscous, incompressible fluid past an isothermal vertical plate in a doubly stratified medium. The governing boundary layer equations are solved numerically using an implicit finite difference scheme of Crank-Nicolson type. The influence of thermophoresis on particle deposition velocity and particle concentration in a doubly stratified medium are analyzed and illustrated graphically. As well the influence of thermal and mass stratification on velocity, temperature and concentration are also investigated and presented. The influence of the parameters on local as well as average skin-friction, the rate of heat and mass transfer are presented graphically and discussed. The results are compared with particular solutions available in the literature and are found to be in good agreement.     

Free surface flow impact on a vertical wall: a numerical assessment

The sudden impact of a free surface flow upon a solid wall is a common occurrence in many situations in nature and technology. The design of marine structures is probably the most obvious example, but also river and dam hydraulics as well as the necessity of understanding flood and debris flow-induced damage have led to theoretical and experimental work on the mechanism of fluid slamming loads. This is therefore a very old and rich research field, which has not yet reached full maturity, so that semi-empirical methods in design practice are still the rule in many sectors. Up-to-date CFD technology with both Eulerian and Lagrangian approaches is employed to investigate highly non-stationary fluid impact on a solid wall. The development of the pressure wave produced by the impact is examined as it propagates and interacts with the fluid boundaries, as well as the subsequent build-up of high-pressure gradients of high fluid velocities. The geometry and the velocity field of the problem considered are very simple, but the results seem to provide new insight, in particular, into the connection between phenomena with different timescales.     

Analysis of the transpiration cooling of a thin porous plate in a hot laminar convective flow

This paper deals with the asymptotic and numerical analysis for the steady-state transpiration cooling of a thin porous flat plate in a laminar hot convective flow, taking into account the streamwise heat conduction through the plate. For high conductivity plates, a regular perturbation analysis has been carried out, yielding a three-term asymptotic solution for the distribution of plate temperature. In the limit of a very poorly conducting plate, a singular perturbation technique, based on matched asymptotic expansions, is employed to solve the governing equations. We also solved the equations numerically using a quasilinearization technique. The numerical results are in good agreement with the asymptotic solution close to the asymptotic limits studied.     

Forced convective flow over a flat plate in non-Newtonian power law fluids

A new forced convection parameter

On the diffusion of a chemically reactive species in a convective flow past a vertical plate

A numerical solution of the first-order homogeneous chemical reaction in an unsteady free convective flow past a semi-infinite vertical plate is studied. The dimensionless governing equations are solved by an efficient, more accurate, unconditionally stable, and rapidly converging implicit finite-difference scheme. The effect of various parameters, such as the Prandtl number, Schmidt number, buoyancy ratio parameter, and chemical reaction parameter on flow velocity and temperature is determined. The velocity profiles are in excellent agreement with available results in the literature. The local and average values of skin friction and Nusselt and Sherwood numbers are calculated. The effects of the chemical reaction parameters on these values are discussed for both generative and destructive reactions. Owing to the presence of the first-order chemical reaction, the velocity is found to increase in the generative reaction and to decrease in the destructive reaction.     

Viscous and Ohmic heating effects in doubly stratified free convective flow over vertical plate with radiation and chemical reaction

An analysis is carried out to study the combined effects of viscous and Ohmic heating in the transient, free convective flow of a viscous, incompressible, and doubly stratified fluid past an isothermal vertical plate with radiation and chemical reactions. The governing boundary layer equations are solved numerically by an implicit finite difference scheme of the Crank-Nicolson type. The influence of different parameters on the velocity, the temperature, the concentration, the skin friction, the Nusselt number, and the Sherwood number is discussed with graphical illustrations. It is observed that an increase in either the thermal stratification or the mass stratification parameter decreases the velocity. An increase in the thermal stratification increases the concentration and decreases the temperature while an opposite effect is observed for an increase in the mass stratification. An augmentation in viscous and Ohmic heating increases the velocity and temperature while decreases the concentration. The results are found to be in good agreement with the existing solutions in literature.     

Free convection from a vertical plate with concentrated and distributed thermal sources

An analysis is developed for the laminar free convection from a vertical plate with uniformly distributed wall heat flux and a concentrated line thermal source embedded at the leading edge. We introduce a parameter=(1 +Q _L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 to describe the relative strength of the two thermal sources; and propose a unified buoyancy parameter=( Ra_L+ Ra_w)^1/5 with=1/(1 +Pr ^–1) to properly scale the dependent and independent variables. The variables are so defined that the resulting nonsimilar boundary-layer equations can describe exactly the buoyancy-induced flow from the dual sources with any relative strength to fluids of any Prandtl number from very small values to infinity. These nonsimilar equations are readily reducible to the self-similar equations of an adiabatic wall plume for=0, and to those of free convection from uniform flux plate for=1. Rigorous finite-difference solutions for fluids of Pr from 0.001 to are obtained over the entire range of from 0 to 1. The effects of both relative source strength and Prandtl number on the velocity profiles, temperature profiles, and the variations of wall temperature, are clearly illustrated.

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Freie Konvektion an einer vertikalen Platte mit einer konzentrierten und einer gleichmäßig verteilten Wärmequelle

Zusammenfassung Für die freie Konvektion an einer vertikalen Platte mit einer gleichmäßig verteilten Wandwärmestromdichte und einer in der Vorderkante eingebetteten linienförmigen Wärmequelle wird eine Berechnungsmethode entwickelt. Zur Beschreibung der relativen Stärke der beiden Wärmequellen führen wir einen Parameter=(1 + Q_L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 ein und schlagen einen vereinheitlichten Auftriebsparameter=( Ra _L+ Ra _w)^1/5 mit=1/(1 +Pr ^–1 für die Skalierung der abhängigen und unabhängigen Variablen vor. Die Variablen werden so definiert, daß mit den sich ergebenden unabhängigen Grenzschichtgleichungen die von den beiden Wärmequellen beliebiger Stärke verursachte Auftriebsströmung von Fluiden beliebiger Prandtl-Zahl genau beschrieben werden kann. Diese unabhängigen Gleichungen können ohne weiteres auf die selbstähnlichen Gleichungen für den Fall einer lokalen Wärmezufuhr an einer sonst adiabatischen Wand für=0 und jenen der freien konvektion an einer Platte mit einheitlichem Wärmestrom für=1 zurückgeführt werden. Für Fluide mit der Prandtl-Zahl zwischen 0,001 und Unendlich werden nach der strengen finite Differenzen-Methode Lösungen im Bereich von zwischen 0 und 1 erhalten. Der jeweilige Einfluß der relativen Quellenstärke und der Prandtl-Zahl auf die Geschwindigkeits- und Temperaturprofile sowie die Veränderung der Wandtemperatur werden deutlich dargestellt.

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Nomenclature C _f friction coefficient - C _p specific heat - f reduced stream function - g gravitational acceleration - k thermal conductivity - L width of the plate - Nu local Nusselt number - Pr Prandtl number - q _w wall heat flux - Q _L heat generated by the line source - Q _w heat released by the uniform-flux wall from 0 tox, q _w Lx - Ra _L local Rayleigh number, g T _L ^* x ³/( ) - Ra _w local Rayleigh number,g T _w ^* w ³/( ) - T fluid temperature - T temperature of ambient fluid - T _L ^* characteristic temperature of the line source,Q _{L/(C p} L) - T _w ^* characteristic temperature of the uniform flux wall, =q _w x/k=Q _w /(C _p L) - u velocity component in then-direction - U₀ dimensionless velocity,u/(/x) Ra _L ^2/5 - U ₁ dimensionless velocity,u/(/x) Ra _w ^2/5 - velocity component in they-direction - x coordinate parallel to the plate - y coordinate normal to the plate - thermal diffusivity - thermal expansion coefficient - pseudo-similarity variable,(y/x) - dimensionless temperature, (T–T )/(T _L ^* +T _w ^* ) - ₀ dimensionless temperature, (Ra_l)^1/5 (T–T )/T _L ^* - ₁ dimensionless temperature, (Ra_w)Ra_w)^1/5 (T–T )/T _w ^* - (Ra _L+Ra_w)^1/5 - kinematic viscosity - (1 +Ra _L/Ra_w)^–1=(1 +T _L ^* /T _w ^* )^–1=(1 + Q_L/Q_w)^–1 - density - Pr/(1 +Pr) - _w wall shear stress - stream function     

Effects of longitudinal heat conduction of a vertical thin plate in a natural convective cooling process

This paper analyzes the cooling process of a vertical thin plate caused by a free convective flow, taking into account the effects of both longitudinal and transversal heat conduction in the plate. Due to the finite thermal conductivity of the plate, a longitudinal temperature gradient arises within it, which prevents any similarity solution in the boundary layer, changing the mathematical character of the problem from parabolic to elliptic, for large values of the Rayleigh number. The energy balance equations are reduced to a system of three differential equations with two parameters: the Prandtl number and a non-dimensional plate thermal conductivity . In order to obtain the evolution of the temperature of the plate as a function of time and position, the coupled balance equations are integrated numerically for several values of the parameters, including the cases of very good and poor conducting plates. The results obtained, are compared with an asymptotic analysis based on the multiple scales technique carried out for the case of a very good conducting plate. There is at the beginning a fast transient in non-dimensional time scale of order ^–1 followed by a slow non-dimensional time scale of order unity, which gives the evolution of the cooling process. Good agreement is achieved even for values of the conduction parameter of order unity. The asymptotic solution allows us to give closed form analytical solution for the plate temperature evolution in time and space. The overall thermal energy of the plate decreases faster for smaller values of .

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Einfluß der Längswärmeleitung in einer senkrechten dünnen Platte auf den Kühlvorgang unter natürlicher Konvektion

Zusammenfassung In dieser Arbeit wird die Abkühlung einer senkrechten dünnen Platte unter freier Konvektion untersucht, wobei die Einflüsse von Längs- und Querwärmeleitung in der Platte Berücksichtigung finden. Aufgrund der endlichen Wärmeleitfähigkeit der Platte bildet sich darin ein Temperaturgradient aus, der Ähnlichkeitslösungen für die Grenzschichtströmung nicht zuläßt, da der mathematische Charakter des Problems für große Werte der Rayleigh-Zahl vom parabolischen in den elliptischen Typ übergeht. Die Energiebilanzgleichungen reduzieren sich auf ein System von drei Differentialgleichungen mit zwei Parametern: die Prandtl-Zahl und eine dimensionslose Temperaturleitfähigkeit des Plattenmaterials. Um die Entwicklung des Temperaturfeldes in der Platte als Funktion von Zeit und Ort verfolgen zu können, werden die gekoppelten Bilanzgleichungen für mehrer Werte der Parameter — einschließlich der Fälle sehr guter und sehr schlechter Wärmeleitfähigkeit — numerisch integriert. Die gefundenen Ergebnisse lassen sich für den Fall der sehr gut leitenden Platte mit den Ergebnissen einer asymptotischen Untersuchung vergleichen. Zu Beginn — in einem dimensionslosen Zeitbereich der Größenordnung ^–1 — zeigt sich ein sehr rasches Übergangsverhalten, gefolgt von einem Zeitbereich der Größenordnung Eins, in dem der eigentliche Kühlungsprozeß abläuft. Selbst für Werte des Leitfähigkeitsparameters der Größenordnung Eins zeigt sich gute Übereinstimmung. Die asymptotische Lösung gibt die Entwicklung des Temperaturfeldes in der Platte nach Zeit und Ort in geschlossener Form wieder. Für kleinere Werte des Parameters nimmt die in der Platte gespeicherte Gesamtenergie schneller ab.

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Nomenclature C _p fluid's specific heat - C _w plate specific heat - f non-dimensional stream function introduced in Eq. (12) - g function introduced in Eq. (19) - G ₀ non-dimensional heat flux given in Eq. (32) - h plate half-thickness - L plate length - Pr Prandtl number - Ra Rayleigh number defined in Eq. (1) - Re Reynolds number - s non-dimensional strained time introduced in Eq. (17) - t time - t _c characteristic time given byt _c= _w C _w h L Ra ^–1/4/ - T temperature - x cartesian lengthwise coordinate - y cartesian transversal coordinate - z non-dimensional transversal coordinate defined in Eq. (3) Greek symbols non-dimensional parameter defined in Eq. (5) - fluid's thermal expansion coefficient - non-dimensional transversal coordinate given in Eq. (11) - _w plate conductivity - fluid's conductivity - non-dimensional transversal coordinate introduced in Eq. (19) - similarity independent variable defined in Eq. (59) - dynamic viscosity - kinematic viscosity - _w plate density - non-dimensional fast time defined in Eq. (17) - non-dimensional time defined in Eq. (3) - _i time adjusted constants - non-dimensional fluid temperature defined in Eq. (11) - _w non-dimensional plate temperature defined in Eq. (3) - x non-dimensional lengthwise coordinate defined in Eq. (3) - non-dimensional self similar independent variable introduced in Eq. (64)     

Non-Darcy effects on convective boundary layer flow past a semi-infinite vertical plate in saturated porous media

The composite effects of viscosity, porosity, buoyancy parameter, thermal conductivity ratio and non-Darcy effects of Brinkman friction and Forscheimmer quadratic drag on the mixed convection boundary layer flow past a semi-infinite plate in a fully-saturated porous regime are theoretically and numerically investigated using Keller’s implicit finite-difference technique and a double-shooting Runge-Kutta method. The Brinkman Forcheimer-extended Darcy model is implemented in the hydrodynamic boundary layer equation. The effects of the various non-dimensional thermofluid parameters, viz Grashof number, Darcy number, and Forchheimer number, and also porosity, thermal conductivity and viscosity parameters on the velocity and temperature fields are discussed. Computations for both numerical schemes are made where possible and found to be in excellent agreement.     

Vortex shedding induced by a solitary wave propagating over a submerged vertical plate

Experimental study was conducted on the vortex shedding process induced by the interaction between a solitary wave and a submerged vertical plate. Particle image velocimetry (PIV) was used for quantitative velocity measurement while a particle tracing technique was used for qualitative flow visualization. Vortices are generated at the tip of each side of the plate. The largest vortices at each side of the plate eventually grow to the size of the water depth. Although the fluid motion under the solitary wave is only translatory, vortices are shed in both the upstream and downstream directions due to the interaction of the generated vortices as well as the vortices with the plate and the bottom. The process can be divided into four phases: the formation of a separated shear layer, the generation and shedding of vortices, the formation of a vertical jet, and the impingement of the jet onto the free surface. Similarity velocity profiles were found both in the separated shear layer and in the vertical jet.     

Mass transfer and free convective MHD flow past an accelerated vertical porous plate

The effects of variable suction/injection on the unsteady two-dimensional free convective flow with mass transfer of an electrically conducting fluid past a vertical accelerated plate in presence of a transverse magnetic field is considered. Solutions of the equations governing the flow are obtained with the help of the power series. The paper is concluded with a discussion of the results obtained.

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Stofftransport und MHD-Strömung bei freier Konvektion an einer beschleunigten senkrechten porösen Platte

Zusammenfassung In dieser Arbeit werden die Wirkungen verÄnderlicher Absaugung/Ausblasung auf die instationÄre zweidimensionale freie Strömung mit Stoffübertragung eines elektrisch leitenden Fluids an einer senkrecht beschleunigten Wand mit magnetischem Querfeld betrachtet. Lösungen erhÄlt man mit Hilfe von PotenzansÄtzen. Die Ergebnisse werden diskutiert.

     

A numerical study on the absorption of water vapor into a film of aqueous LiBr falling along a vertical plate

Absorber is an important component in absorption machines and its characteristics have significant effects on the overall efficiency of absorption machines. This article reports a model of simultaneous heat and mass transfer process in absorption of refrigerant vapor into a lithium bromide solution of water––cooled vertical plate absorber in the Reynolds number range of 5 < Re < 150. The boundary layer assumptions were used for the transport of mass, momentum and energy equations and the fully implicit finite difference method was employed to solve the governing equations in the film flow. Dependence of lithium bromide aqueous properties to the temperature and concentration and film thickness to vapor absorption was employed. This model can predict temperature, concentration and properties of aqueous profiles as well as the absorption heat and mass fluxes, heat and mass transfer coefficients, Nusslet and Sherwood number of absorber. An analysis for linear distribution of wall temperature condition carries out to investigation the reliability of the present numerical method through comparing with previous investigation.     

The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder

This paper deals with a theoretical (numerical) analysis of the effects that blowing/injection and suction have on the steady mixed convection or combined forced and free convection boundary layer flows over a vertical slender cylinder with a mainstream velocity and a wall surface temperature proportional to the axial distance along the surface of the cylinder. Both cases of buoyancy forces aid and oppose the development of the boundary layer are considered. Similarity equations are derived and their solutions are dependent upon the mixed convection parameter, the non-dimensional transpiration parameter and the curvature parameter, as well as of the Prandtl number. Dual solutions for the previously studied mixed convection boundary layer flows over an impermeable surface of the cylinder are shown to exist also in the present problem for aiding and opposing flow situations.     

Combined forced and free convective flow of water at 4°C past a semi-infinite vertical plate

Effects of buoyancy forces on forced and free convective flow of water at 4°C past a semi-infinite vertical plate at constant temperature are studied. Flow is assumed to be vertically upwards. Similarity solutions are derived and the resulting equations are solved numerically on a computer. Velocity and temperature profiles are shown graphically and numerical values of the skin friction and the rate of heat transfer are entered in tables. It is observed that the skin friction and the Nusselt number increase with increasing Gr/Re², where Gr is the Grashof number and Re is the Reynolds number     

Experimental study on convective heat transfer coefficient around a vertical hexagonal rod bundle

Research on convective heat transfer coefficient around a rod bundle has many diverse applications in industry. So far, many studies have been conducted in correlations related to internal and turbulent fully-developed flow. Comparison shows that Dittus-Boelter, Sieder-Tate and Petukhov have so far been the most practical correlations in fully-developed turbulent fluid flow heat transfer. The present study conducts an experimental examination of the validity of these frequently-applied correlations and introduces a manufactured test facility as well. Due to its generalizibility, the unique geometry of this test facility (hexagonal arranged, 7 vertical rods in a hexagonal tube) can fulfil extensive applications. The paper also studies the major deviation sources in data measurements, calibrations and turbulence of fluid flow in this. Finally, regarding to sufficient number of experiments in a vast fluid mean velocity range (3,800?<?Re?<?40,000), a new curve and correlation are presented and the results are compared with the above mentioned commonly-applied correlations.