E. Zanchini

On the choice of the reference temperature for fully-developed mixed convection in a vertical channel

Abstract The effect of the choice of the reference fluid temperature on the solutions of fully-developed mixed-convection problems in a plane vertical channel is studied. First, the boundary conditions of either uniform wall temperatures or a uniform temperature at a wall and a uniform heat flux on the opposite wall are considered. It is shown that, in these cases, the choice of the reference temperature affects both the velocity profiles and the axial change of the difference between the pressure and the hydrostatic pressure. A general method to choose the reference fluid temperature for the fully developed mixed convection in ducts is proposed. Finally, an analytical solution for the boundary condition of uniform wall heat fluxes is obtained by choosing the mean fluid temperature in each cross section as the local reference temperature.

Effect of viscous dissipation on mixed convection in a vertical channel with boundary conditions of the third kind

Abstract The effect of viscous dissipation on fully-developed mixed convection is analysed for the laminar flow in a parallel-plate vertical channel whose walls exchange heat with an external fluid. Both conditions of equal and of different reference temperatures of the external fluid are considered. First, the simpler cases of either negligible Brinkman number or negligible Grashof number are solved analytically. Then, the combined effects of buoyancy forces and of viscous dissipation are analysed by a perturbation series method. In the examined cases, the velocity field, the temperature field and the Nusselt numbers are evaluated.

Effect of viscous dissipation on the asymptotic behaviour of laminar forced convection in circular tubes

Abstract The asymptotic behaviour of laminar forced convection in a circular tube, for a Newtonian fluid at constant properties, is analysed by taking into account the viscous dissipation effects. The axial heat conduction in the fluid is neglected. A sufficient condition for the existence of a fully developed region is determined. This condition includes, for instance, any asymptotically vanishing axial distribution of the wall heat flux, uniform wall temperature, convection with an external fluid. The asymptotic temperature field and the asymptotic value of the Nusselt number are determined analytically, for every boundary condition which allows a fully developed region. In particular, it is proved that, whenever the wall heat flux tends to zero, the asymptotic Nusselt number is zero. Copyright

Forced convection in the thermal entrance region of a circular duct with slug flow and viscous dissipation

Abstract Slug flow forced convection in a circular duct is studied. The effect of viscous dissipation is analysed in the thermal entrance region. The temperature field and the local Nusselt number are determined analytically for any prescribed axial distribution of wall heat flux. Three examples are considered: a uniform wall heat flux, a linearly varying wall heat flux and an exponentially varying wall heat flux. In the case of a uniform wall heat flux, it is shown that viscous dissipation reduces the value of the local Nusselt number in the whole duct. In the case of a linearly or exponentially increasing wall heat flux, viscous dissipation affects the local Nusselt number only in the thermal entrance region and becomes negligible in the fully developed region.

Finite-difference solution of hyperbolic heat conduction with temperature-dependent properties

Abstract A numerical evaluation of the temperature field in an infinite solid medium that surrounds a cylindrical surface is presented. An unsteady and uniform heat flux density is prescribed at the cylindrical surface, and Cattaneo-Vernottes constitutive equation for the heat flux density is supposed to hold. The hyperbolic differential problem is solved by MacCormacks predictor-corrector method by assuming that both the thermal conductivity and the specific heat are temperature-dependent. Then, the results of the numerical evaluation are compared with the analytical solution that is available in the literature for the special case of constant thermophysical properties.

Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady-periodic electric field

Abstract Hyperbolic heat conduction in an infinitely long cylindrical solid with internal heat generation produced by Joule effect is considered. The power generated per unit volume is non-uniform and steady periodic. The surface of the cylinder is assumed to exchange heat by convection with an external fluid. The temperature field within the cylinder is determined analytically in a steady-periodic regime. For a fixed material and for a fixed radius of the cylinder, the dependence of the amplitude of thermal waves on the frequency of the electric current is studied and it is proved that thermal resonances occur.

Time-periodic laminar mixed convection in an inclined channel

Abstract The steady-periodic regime of laminar mixed convection in an inclined channel is studied analytically, with the following boundary conditions: the temperature of one channel wall is stationary, while the temperature of the other wall is a sinusoidal function of time. Analytical expressions of the velocity field, of the temperature field, of the pressure drop, of the friction factors, as well as of the Nusselt number at any plane parallel to the walls are determined. It is found that, for every value of the Prandtl number greater than 0.277, there exists a resonance frequency which maximizes the amplitude of the friction factor oscillations at the unsteady-temperature wall. Moreover, for any plane which lies between the midplane of the channel and the unsteady-temperature wall, every value of the Prandtl number yields a resonance frequency which maximizes the amplitude of the Nusselt number oscillations.

Mixed convection with viscous dissipation in an inclined channel with prescribed wall temperatures

Abstract The fully developed laminar mixed convection with viscous dissipation in an inclined channel with prescribed wall temperatures is studied analytically. The mean fluid temperature is assumed as the reference temperature. Two perturbation expansions are considered. In the first, the forced convection with viscous dissipation is assumed as a starting condition and the effects of buoyancy for fixed values of the Brinkman number are studied. In the second, starting from the solution for mixed convection without viscous dissipation, the effects of the Brinkman number for fixed values of the Grashof number are analysed. The different solution methods allow a cross-check of the results. The dimensionless velocity field, the dimensionless temperature field, the dimensionless pressure field, the friction factors and the Nusselt numbers are determined and discussed. The results show that viscous dissipation enhances the effects of buoyancy and vice versa.

Laminar forced convection with sinusoidal wall heat flux distribution: axially periodic regime

Stationary and laminar forced convection in a circular tube with a sinusoidal axial distribution of wall heat flux is studied under the hypothesis that both axial heat conduction and viscous dissipation in the fluid are negligible. Two cases are considered: a sinusoidal wall heat flux distribution with a vanishing mean value; a sinusoidal wall heat flux distribution which does not change its sign. In both cases, the temperature field and the local Nusselt number are evaluated analytically in the fully developed region, i.e. where the local Nusselt number depends periodically on the axial coordinate. It is shown that, in the first case, the fully developed region presents an infinite sequence of axial positions where the local Nusselt number is singular. In these positions, the wall heat flux has a non-vanishing value even if the wall temperature equals the bulk temperature.ZusammenfassungUnter Vernachlässingung der Längswärmeleitung und der Dissipationsenergie wird die stationäre, laminare Zwangskonvektion in einem Kreisrohr mit axial sinusförming veränderlichen Wandwärmefluß untersucht, und zwar für zwei Fälle: Sinusverteilung mit verschwindendem Mittelwert einerseits und ohne Vorzeichenwechsel andererseits. In beiden Fällen werden das Temperaturfeld und die lokale Nusselt-Zahl für den „vollausgebildeten Bereich” analytisch ermittelt (die lokale Nusselt-Zahl ist dort eine periodische Funktion der Längskoordinate). Es wird gezeigt, daß im ersten Fall innerhalb des vollausgebildeten Bereiches eine endliche Folge von singulären Stellen bezüglich der lokalen Nusselt-Zahl existiert, d.h. trotz Gleichheit von Wand- und Mischtemperatur behält der aufgeprägte Wärmefluß dort einen endlichen Wert.

Three-dimensional propagation of hyperbolic thermal waves in a solid bar with rectangular cross-section

Abstract The hyperbolic heat conduction in a solid bar with a rectangular section is analyzed. Each surface of the bar is subjected to a uniform and time varying heat flux. It is shown that, according to the heat-flux formulation of hyperbolic heat conduction, the heat flux density field can be determined by employing the analytical solution of a one-dimensional problem. The distributions of the heat flux density and the temperature are obtained for a bar with a finite length and with arbitrary time-evolutions of the heat flux on its surfaces. Special attention is devoted to a two-dimensional case, i.e. that of a bar with insulated ends. In this case, plots of the temperature field at given instants of time are reported and compared with those which correspond to a vanishing relaxation time.