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)     

Radiation effect on forced convective flow and heat transfer over a porous plate in a porous medium

Heat transfer analysis has been presented for the boundary layer forced convective flow of an incompressible fluid past a plate embedded in a porous medium. The similarity solutions for the problem are obtained and the reduced nonlinear ordinary differential equations are solved numerically. In case of porous plate, fluid velocity increases for increasing values of suction parameter whereas due to injection, fluid velocity is noticed to decrease. The non-dimensional temperature increases with the increasing values of injection parameter. A novel result of this investigation is that the flow separation occurred due to suction/injection may be controlled by increasing the permeability parameter of the medium. The effect of thermal radiation on temperature field is also analyzed.     

Steady laminar flow past a heated horizontal plate embedded in a saturated porous medium

In this paper we study the boundary layer equations for steady laminar flow past a heated horizontal plate embedded in a saturated porous medium by adopting the formulation of Chandrasekhara [3], Kolar and Sastri [7]. The velocity distribution and temperature distribution are determined by using the implicit Crank-Nicolson-Predictor-Corrector method of finite difference scheme [7] and [1]. With the help of a compute the distributions are estimated at both (i+1/2)th and (i+1)th levels and they are presented in tabular form. The curves for these distributions are plotted. We calculate the shear stress and skin friction at the wall and observe that the skin-friction directly depends upon the dimensions of the plate and inversely depends upon the Reynolds numberRe. The heat flux and the Nusselt number are evaluated. Further we observe that the Nusselt number depends upon the length of the porous plate.

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Stetige laminare Strömung über eine in einem gesättigten porösen Medium eingebettete horizontale Platte

Zusammenfassung Diese Untersuchung befaßt sich mit den Grenzschichtgleichungen für eine stetige laminare Strömung über eine beheizte Platte, die in ein gesättigtes poröses Medium eingebettet ist, mittels des Formalismus von Chandrasekhara [3], Kolar und Sastri [7]. Die Geschwindigkeits- und Temperaturverteilung wurden unter Benutzung der impliziten Crank-Nicolson-Korrekturmethode des Finiten-Elemente-Schemas bestimmt. Die (i+1/2). und die (i+1). Ebene der Verteilungen wurden mit Computer-Hilfe berechnet und in Tabellenform dargestellt. Die Graphen der Temperatur- und Geschwindigkeitsverteilung wurden ausgeplottet. Die Schubspannungen und die Oberflächenreibung an der Wand wurden berechnet und es konnte festgestellt werden, daß die Oberflächenreibung direkt von der Größe der Platte abhängt und umgekehrt proportional der Reynolds-ZahlRe ist. Der Wärmestrom und die Nusselt-Zahl wurden bestimmt. Weiterhin konnte festgestellt werden, daß die Nusselt-Zahl von der Länge der porösen Platte abhängt.

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Nomenclature C _p specific heat of the convective fluid - D skin friction - k permeability of the porous medium - k _f thermal conductivity of the fluid - k _m the coefficient of thermal conductivity of the porous medium - k _s the conductivity of the solid matrix - N(x) Nusselt number - q(x) specific heat flux - Re local Reynolds number - T temperature - T ₀ temperature of the free stream - T _w temperature of the plate - u velocity in thex-direction - u ₀ velocity of the free stream - V velocity iny-direction - x coordinate axis along the plate - y coordinate axis normal to the plate Greek symbols thermal diffusivity - thickness of the velocity boundary layers in thex direction - thickness of the velocity boundary layer in they-direction - the porosity of the medium - dimensionless variable - kinematic viscosity of the fluid - density of the fluid - shear stress     

Free convective transpiration over a vertical plate: a numerical study

A rarely adopted simple finite difference scheme has been successfully employed to solve the nonlinear coupled partial differential equations, with nonhomogeneous boundary condition, which describe the free convection at a vertical plate with transpiration. The solution is obtained for a Prandtl number of 0.72, in the blowing parameter range of — 1.9 < C_x < 1.9. The effects of suction and blowing on heat transfer and skin friction are discussed. It is concluded that the boundary layer has a better memory of the upstream suction distribution than of the upstream blowing distribution.     

Numerical simulation of transpiration cooling through porous material

Transpiration cooling using ceramic matrix composite materials is an innovative concept for cooling rocket thrust chambers. The coolant (air) is driven through the porous material by a pressure difference between the coolant reservoir and the turbulent hot gas flow. The effectiveness of such cooling strategies relies on a proper choice of the involved process parameters such as injection pressure, blowing ratios, and material structure parameters, to name only a few. In view of the limited experimental access to the subtle processes occurring at the interface between hot gas flow and porous medium, reliable and accurate simulations become an increasingly important design tool. In order to facilitate such numerical simulations for a carbon/carbon material mounted in the side wall of a hot gas channel that are able to capture a spatially varying interplay between the hot gas flow and the coolant at the interface, we formulate a model for the porous medium flow of Darcy–Forchheimer type. A finite‐element solver for the corresponding porous medium flow is presented and coupled with a finite‐volume solver for the compressible Reynolds‐averaged Navier–Stokes equations. The two‐dimensional and three‐dimensional results at Mach number Ma = 0.5 and hot gas temperature T_HG=540 K for different blowing ratios are compared with experimental data. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

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.

     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Dynamic and convective results for a developing laminar unsteady flow

Dynamic and thermal results for developing laminar pulsed flows in a duct are presented. They have been investigated by means of a finite difference model. This flow is described in terms of an unsteady pulsed flow superimposed on a steady incompressible one with the following main assumptions: a sinusoidal modulation for the pulsation and a uniform wall temperature. Results emphasize the importance of this entry region, where four simultaneous developments occur: steady—dynamic and thermal—and unsteady—dynamic and thermal.     

Magnetohydrodynamics convective flow in a vertical slot through a porous medium

An analysis is presented with magnetohydrodynamics natural convective flow of a viscous Newtonian fluid saturated porous medium in a vertical slot. The flow in the porous media has been modeled using the Brinkman model. The fully-developed two-dimensional flow from capped to open ends is considered for which a continuum of solutions is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. The asymptotic behaviour and the volume flux are analyzed and incorporated graphically for the three-parameter family of solution.     

An experimental study on the efficiency of transpiration cooling in laminar and turbulent hypersonic flows

An experimental study on the efficiency of transpiration cooling in hypersonic laminar and turbulent flow regimes is carried out in the Hypersonic Windtunnel Cologne with a focus on the aerothermal problems downstream of the cooled model part. The model is made of a material of low thermal conductivity (PEEK) with an integrated probe of a porous material. The experimental setup allows the direct comparison of the thermal behavior of transpiration cooling to a well-defined and radiatively cooled reference surface. Experiments are performed at Mach number of 6 and two different Reynolds numbers. Air, argon and helium are used as coolants at various flow rates, in order to identify the influence of coolant medium on cooling efficiency. The cooling efficiency of air and argon is comparable. Helium provides significantly higher cooling efficiency at the same blowing ratio, i.e. same coolant mass flow rate. The experimental data shows that the efficiency of the transpiration cooling in turbulent flows is much lower than in laminar flow.     

Numerical predictions of the laminar flow over a normal flat plate

Finite-difference and finite-element techniques have been used to calculate the steady laminar flow over a flat plate normal to an air stream, up to a Reynolds number, Re, based on the plate half-width, of 100. The boundary conditions simulate a central splitter plate downstream of the body, to prevent vortex shedding, so the flow is characterized by a closed recirculation region which grows with increasing Re but at Re = O(100) is very similar in size to the turbulent recirculating region that occurs in the corresponding high Reynolds-number flow. Motivation came, in part, from the increasing efforts of turbulence modellers to calculate complex turbulent flows (containing elliptic regions) and our belief that the numerical methods commonly employed for such work can be inaccurate. The predictions are compared with each other and with some expectations based on classic solutions of the Navier-Stokes equations, and the nature of the numerical errors is demonstrated. It is concluded that effort comparable with that expended in developing turbulence models should be directed to developing higher-order numerical methods, before the numerical accuracy of predictions of, for example, bluff-body flows can be made sufficiently high to sustain detailed discussion of the adequacy of turbulence models in such situations.     

Numerical analysis of flow pattern of impinging liquid sprays in a cold model for cooling a hot flat plate

This paper treats the numerical analysis of two-phase mist jet flow, which is commonly adopted to cool the solidified shell in the secondary cooling zone of the continuous casting process. Flow structures of the two-phase subsonic jet impinging on a flat plate normal to flow, corresponding to the present cooling situation, are solved on the assumption that particles are perfectly elastically reflected from a surface. Again, the numerical experiments concerning mist flows composed of air and water-droplets are made in a cold model. The flow fields for both gas and particle phases strongly depend upon the particle size. When waterdroplets mixing in the mist are very small, the impinging particles travel very closely to the surface. With increasing particle size, particles are reflected from the surface in a far distance. Therefore, also, the case is analysed where a low velocity annular gas-only flow surrounding a round nozzle co-axially is present so that such idle particles may be pushed back to the surface again. This is considered to result in an improvement of the mist cooling efficiency.     

Local convective heat transfer for laminar and turbulent flow in a rotor-stator system

Abstract The aim of this work is to show a better comprehension of the flow structure and thermal transfer in a rotor-stator system with a central opening in the stator and without an airflow imposed. The experimental technique uses infrared thermography to measure the surface temperatures of the rotor and the numerical solution of the steady-state heat equation to determine the local heat transfer coefficients. Analysis of the flow structure between the rotor and the stator is conducted by PIV. Tests are carried out for rotational Reynolds numbers ranging from 5.87×10⁴ to 1.4×10⁶ and for gap ratios ranging from 0.01 to 0.17. Analysis of the experimental results has determined the influence of the rotational Reynolds number, the gap ratio and systems geometry on the flow structure, and the convective exchanges in the gap between the rotor and the stator. Some correlations expressing the local Nusselt number as a function of the rotational Reynolds number and the gap ratio are proposed.     

Analysis of a laminar boundary layer flow over a flat plate with injection or suction

An analysis is performed to study a laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. The basic equations of this problem are reduced to a system of nonlinear ordinary differential equations by means of appropriate transformations. These equations are solved analytically by the optimal homotopy asymptotic method (OHAM), and the solutions are compared with the numerical solution (NS). The effect of uniform suction/injection on the heat transfer and velocity profile is discussed. A constant surface temperature in thermal boundary conditions is used for the horizontal flat plate.     

Stability of stationary convective flow of a mixture in a vertical porous layer

In contrast to the corresponding viscous flow, the convective flow of a homogeneous liquid in a planar vertical layer whose boundaries are maintained at different temperatures is stable [1]. When a porous layer is saturated with a binary mixture, in the presence of potentially stable stratification one must expect an instability of thermal-concentration nature to be manifested. This instability mechanism is associated with the difference between the temperature and concentration relaxation times, which leads to a buoyancy force when an element of the fluid is displaced horizontally. In viscous binary mixtures, the thermal-concentration instability is the origin of the formation of layered flows, which have been studied in detail in recent years [2–4]. The convective instability of the equilibrium of a binary mixture in a porous medium was considered earlier by the present authors in [5]. In the present paper, the stability of stationary convective flow of a binary mixture in a planar vertical porous layer is studied. It is shown that in the presence of sufficient longitudinal stratification the flow becomes unstable against thermal-concentration perturbations; the stability boundary is determined as a function of the parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–157, January–February, 1980.