Laminar natural convection over a slender vertical frustrum of a cone with constant wall heat flux

The problem of laminar natural convection flow over a slender frustrum of a cone with constant wall heat flux is treated in this paper. The governing differential equations are solved by a combination of quasilinearization and finite-difference methods. Numerical solutions are obtained for Prandtl numbers from 0.1 to 100 for a range of values of transverse curvature parameter. It is found that the effect of transverse curvature is of great significance in such flows.

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Laminare natürliche Konvektion über einem dünnen, senkrechten Kegelstumpf mit konstantem Wand wär mestrom

Zusammenfassung In dieser Arbeit wird das Problem der laminaren, natürlichen Konvektionsströmung öber einem dünnen Kegelstumpf mit konstantem Wandwärmestrom behandelt. Die maßgeblichen Differentialgleichungen werden mit Hilfe einer Kombination von Quasilinearisierung und Differenzenverfahren gelöst. Numerische Lösungen werden für die Prandtl ' sehen Zahlen zwischen 0. l und 100 innerhalb eines Bereiches von Querkrüm mungswerten erhalten. Es wird gezeigt, daß der Einfluß der Querkrümmung in solchen Strömungen von großer Bedeutung ist.

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Nomenclature A,B,C constants in the transformation, defined in Eq.(14) - f dependent variable, defined in Eq. (7) - g dependent variable, defined in Eq. (7) - g_e gravitational acceleration - k heat conductivity - k_n -grid - L characteristic length - Nu Nusselt number - Pr Prandtl number - q_w wall heat flux - r radial distance from the axis of the cone - R_TVC transverse curvature ratio, defined in Eq.(28) - Re Reynolds number - T temperature - u,v velocity components in the x- and y-directions, respectively - x,y rectangular coordinates Greek Letters dimensionless temperature, defined in Eq.(4) - bulk modulus - cone angle - dynamic viscosity - stream function - , transformed independent variables, defined in Eq. (7) - transverse curvature parameter     

Natural convection flow over a vertical frustum of a cone for constant wall heat flux

Laminar natural convection flow and heat transfer over a vertical frustum of a cone has been studied. The governing boundary layer equations are solved using local non-similarity method for constant wall heat flux. The local similarity and the local non-similarity two and three-equation models are constructed and the resulting equations are solved numerically. Results obtained from two and three-equation models are in good agreement. The numerical values of the flow and temperature functions required to calculate the surface skin friction and heat transfer rate are reported for various values of Prandtl numbers.     

Non-similar solutions of free convection flow over a vertical frustum of a cone for constant wall temperature

Summary The laminar free convection flow and heat transfer over a vertical frustum of a cone is studied. The governing boundary layer equations are solved using local non-similarity method for constant wall temperature case. Local similarity and the local non-similarity two- and three-equation models are constructed and the resulting equations are solved numerically. Results obtained from two- and three-equation models are in good agreement. The numerical values of the flow and temperature functions required to calculate the surface skin friction and heat transfer rate have been reported for various values of Prandtl numbers.

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Nichtähnlichkeits-Lösnngen für die freie Konvektionsströmung an einem vertikalen Kegelstumpf bei konstanter Wandtemperatur

Übersicht Untersucht wird die laminare freie Konvektionsströmung und der Wärmeübergang an einem vertikalen Kegelstumpf. Zur Lösung der zugehörigen Grenzschichtgleichungen im Fall konstanter Wandtemperatur werden die Modelle der lokalen Ähnlichkeit und der lokalen Nichtähnlichkeit mit zwei bzw. drei Gleichungen eingeführt und die sich ergebenden Gleichungen numerisch gelöst. Die Lösungen nach den Modellen mit zwei und drei Gleichungen passen gut zusammen. Die Zahlenwerte von Strom- und Temperaturfunktionen, die für die Berechnung der Wandschubspannung und der Wärmeübergangsrate benötigt werden, sind für verschiedene Werte der Prandtl-Zahl angegeben.

     

Turbulent natural convection over a slender circular cylinder

The transverse-curvature effect on the heat transfer in the turbulent natural convection flow from the outer surface of a slender vertical circular cylinder is studied by an improved integral method for various values of Prandtl numbers and for various values of a transverse curvature parameter.     

Laminar natural convection between partially heated vertical parallel plates

A numerical analysis has been performed to examine the characteristics of laminar natural convection in vertical channel with unheated entry and unheated exit. The heated section is subjected to uniform wall temperature (UWT) or uniform heat flux (UHF). Theoretical results for average Nusselt number and induced volume flow rateQ were derived under fully developed condition. Particular attention is paid to investigating the effects of the partially heated section on the induced volume flow rate and Nusselt number for various conditions. Results show that for UWT the induced volume flow rateQ increases with decreasing unheated entry length or increasing total length of heated section and unheated exit. For a fixed unheated entry length, the channel with a longer heated section length causes a greaterQ. Additionally, for both UWT and UHF, the average Nusselt number under fully-developed condition increases with increasing value ofE ₁/E ₂.

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Laminare Naturkonvektion zwischen teilbeheizten senkrechten Parallelplatten

Zusammenfassung Die numerische Untersuchung soll das Verhalten einer laminaren Strömung bei natürlicher Konvektion in einem senkrechten Kanal mit unbeheiztem Ein- und Austritt klären. Der beheizte Abschnitt wird entweder mit gleichförmiger Wandtemperatur (UWT) oder gleichförmigem Wärmefluß (UHF) beaufschlagt. Bezüglich voll ausgebildeter Strömung ließen sich theoretische Ergebnisse für die mittlere Nußelt-Zahl und den induzierten VolumenstromQ gewinnen. Besonderes Interesse galt der Untersuchung des Einflusses des teilbeaufschlagten Abschnittes auf Volumenstrom und Nußelt-Zahl unter verschiedenen Nebenbedingungen. Die Ergebnisse zeigen, daß im UWT-Fall der induzierte VolumenstromQ mit abnehmender unbeheizter Einlauflänge oder zunehmender Gesamtlänge des Heizabschnittes anwächst. Bei fester unbeheizter Einlauflänge erzeugt der Kanal mit längerem Heizabschnitt einen höheren StromQ.

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Nomenclature b half channel width - E ₁ ratio of unheated exit length to channel length,l ₁/l - E ₂ ratio of heated section length to channel length,l ₂/l - E ₃ ratio of unheated entry length to channel length,l ₃/l - Gr _L Grashof number, Eq. (4) - g gravitational acceleration - k thermal conductivity - l channel length - l ₁ unheated exit length - l ₂ heated section length - l ₃ unheated entry length - L dimensionless channel length, Eq. (4) - L ₁ dimensionless unheated exit length - L ₂ dimensionless heated section length - L ₃ dimensionless unheated entry length - average Nusselt number - Nu _x local Nusselt number - p pressure defect - P dimensionless pressure defect, Eq. (4) - Pr Prandtl number,/ - q _w wall heat flux of heated section - Q dimensionless induced volume flow rate, Eq. (8) - Ra _E Rayleigh number based on the heated section length,Ra _L/E₂ - Ra _L Rayleigh number based on the full channel length,Gr _LPr - T temperature - T ₀ inlet temperature - T _w wall temperature - u, v velocity components in thex andy directions, respectively - U, V dimensionless velocity components in thex andy directions, respectively, Eq. (4) - u ₀,U ₀ dimensional and dimensionless inlet velocity, respectively - x, y coordinates in thex andy directions, respectively - X, Y dimensionless coordinates in thex andy directions, respectively, Eq. (4) - X ratio of longitudinal distance from the entrance of heated section to the heated section length,X=[X–(L–L ₁–L ₂)]/L ₂ Greek symbols thermal diffusivity - thermal expansion coefficient - kinematic viscosity - dimensionless temperature, Eq. (4) - ₀ fluid density at ambient temperature     

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.     

Laminar natural convection of pure and saline water along a vertical isothermal cylinder

 In most studies concerning laminar natural convection along a vertical isothermal cylinder a linear relationship between fluid density and temperature has been used and kinematic viscosity and thermal diffusivity have been considered constant calculated at ambient temperature. However, it is known that the density–temperature relationship for water is non-linear at low temperatures and kinematic viscosity and thermal diffusivity are functions of temperature. In this study the problem of laminar natural convection of pure and saline water along a vertical isothermal cylinder has been investigated in the temperature range between 20 and 0 ^∘C taking into account the temperature dependence of ν, α and ρ. The results are obtained with the numerical solution of the boundary layer equations. The variation of ν, α and ρ with temperature has a strong influence on free convection characteristics. Received on 17 May 1999     

Conjugate free convection over a vertical slender hollow cylinder embedded in a porous medium

A numerical study of the steady conjugate free convection over a vertical slender, hollow circular cylinder with the inner surface at a constant temperature and embedded in a porous medium is reported. The governing boundary layer equations for the fluid-saturated porous medium over the cylinder along with the one-dimensional heat conduction equation for the cylinder are cast into dimensionless form, by using a non-similarity transformation. The resulting non-similarity equations with their corresponding boundary conditions are solved by using the Keller box method. Emphasis is placed on the effects caused by the wall conduction parameter, p, and calculations have covered a wide range of this parameter. Heat transfer results including the temperature profiles, the interface temperature profiles and the local Nusselt number are presented. Received on 17 November 1997     

Laminar natural convection heat transfer from vertical plate to power-law fluid

Summary Approximate solutions for laminar natural convection heat transfer between a vertical plate and a power-law fluid with high Prandtl number were obtained using an integral method for cases with various types of boundary conditions. The results were found in good agreement with available experimental evidence.Nomenclature a exponent defined by equations (28) and (29) - A, B, C, D, E constants defined by equations (15) to (19) - C ₁, C ₂, M ₁, M ₂ coefficients for Nusselt number expression defined by (32b), (33b) - f temperature difference, equal to T _s–T - f ⁺ dimensionless temperature difference - g gravitational acceleration - Gr Grashof number defined by (25), (50) and (66), respectively - H heat flux at plate surface - h _x local heat transfer coefficient - K consistency index for Power-law fluid - k thermal conductivity of fluid - K ₁, K ₂ constants defined by (50) and (51) - L height of plate - n flow behavior index for Power-law fluid - P a quantity defined by (54a) - T temperature - T _s plate temperature - T temperature of the bulk of fluid - s constant given by (35) - u velocity component along x-direction - u _x maximum velocity induced by natural convection current, (10) - v velocity component along y-direction - x distance measured along direction parallel to that of gravitational force - x ⁺ dimensionless quantity, defined as x/L - y distance measured away from plate - Nu _x local Nusselt number - Nu _av average Nusselt number - Pr Prandtl number defined by (24) - T temperature difference according to boundary conditions - thermal diffusivity of fluid - coefficient of thermal expression of fluid - boundary layer thickness - ⁺ dimensionless boundary layer thickness - dimensionless velocity profile - dimensionless variable, defined as y/ - dimensionless temperature difference     

Transient natural convection flow of a nanofluid over a vertical cylinder

An analysis is performed to study unsteady free convective boundary layer flow of a nanofluid over a vertical cylinder. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing equations are formulated and a numerical solution is obtained by using an explicit finite-difference scheme of the Crank-Nicolson type. The solutions at each time step have been found to reach the steady state solution properly. Numerical results for the steady-state velocity, temperature and nanoparticles volume fraction profiles as well as the axial distributions and the time histories of the skin-friction coefficient, Nusselt number and the Sherwood number are presented graphically and discussed.     

Laminar MHD natural convection of nanofluid containing gyrotactic microorganisms over vertical wavy surface saturated non-Darcian porous media

Magnetohydrodynamic (MHD) bioconvection of an incompressible electrically conducting nanofluid near a vertical wavy surface saturated porous medium containing both nanoparticle and gyrotactic microorganisms is investigated. The nanofluid is represented by a model that includes both Brownian motion and thermophoresis effects. A suitable set of non-dimensional variables are used to transform the governing boundary layer equations into a dimensionless form. The resulting nonlinear system is mapped to the vertical flat plate domain, and a non-similar solution is used to the obtained equations. The obtained non-similar system is then solved numerically using the fourth-order Runge-Kutta method. The influence of various physical parameters on the local Nusselt number, the local Sherwood number, the local density number of the motile microorganisms, the dimensionless velocity, the dimensionless temperature, and the rescaled density of motile microorganisms is studied. It is found that the local Nusselt number, the local Sherwood number, and the local density number of the motile microorganisms decrease by increasing either the Grashof number or the magnetic field parameter.     

A transient natural convection of micropolar fluids over a vertical cylinder

A transient free convective boundary layer flow of micropolar fluids past a semi-infinite cylinder is analysed in the present study. The transformed dimensionless governing equations for the flow, microrotation and heat transfer are solved by using the implicit scheme. For the validation of the current numerical method heat transfer results for a Newtonian fluid case where the vortex viscosity is zero are compared with those available in the existing literature, and an excellent agreement is obtained. The obtained results concerning velocity, microrotation and temperature across the boundary layer are illustrated graphically for different values of various parameters and the dependence of the flow and temperature fields on these parameters is discussed. An increase in the vortex viscosity tends to increase the magnitude of microrotation and thus decreases the peak velocity of fluid flow. An increase in the vortex viscosity in micropolar fluids is shown to decrease the heat transfer rate.     

Modeling natural convection boundary layer flow of micropolar nanofluid over vertical permeable cone with variable wall temperature

This paper discusses the natural convection boundary layer flow of a micropolar nanofluid over a vertical permeable cone with variable wall temperatures. Non-similar solutions are obtained. The nonlinearly coupled differential equations under the boundary layer approximations governing the flow are solved numerically using an efficient, iterative, tri-diagonal, implicit finite difference method. Different experimental correlations for both nanofluid effective viscosity and nanofluid thermal conductivity are considered.It is found that as the vortex-viscosity parameter increases, both the velocity profiles and the local Nusselt number decrease. Also, among all the nanoparticles considered in this investigation, Cu gives a good convection.     

Combined convection on a vertical slender cylinder in a micropolar fluid

An analysis is presented for the steady state mixed convective boundary layer flow of a micropolar fluid along vertical slender cylinders. The governing equations have been solved numerically. Results for the friction factor, Nusselt number as well as the details of flow and temperature fields are displayed for a range of values of the transverse curvature and material parameters for the micropolar fluid. It is observed that micropolar fluids display drag reduction as well as heat transfer rate reduction when compared to Newtonian fluids.

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Mischkonvektion an einem senkrechten schlanken Zylinder in einem mikropolaren Fluid

Zusammenfassung Die Untersuchung bezieht sich auf stationäre Mischkonvektion in der Grenzschicht einer mikropolaren Flüssigkeit entlang eines senkrechten schlanken Zylinders. Die bestimmenden Gleichungen wurden numerisch gelöst. In einem gewissen Bereich des Krümmungsverhältnisses und der Stoffparameter des mikropolaren Fluids werden Ergebnisse für den Widerstandsbeiwert, die Nusselt-Zahl sowie Besonderheiten des Strömungs- und Temperaturfeldes mitgeteilt. Wie sich zeigte, weisen mikropolare Fluide gegenüber Newtonschen Fluiden sowohl geringeren Widerstand als auch niedrigere Wärmeübertragungsintensität auf.

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