Transient heat conduction with uniform heat generation in a slab subjected to convection and radiation cooling

A finite difference scheme with fourth order Runge-Kutta method is employed to determine the unsteady state temperature distribution in a plane slab with uniform heat generation. The plane slab is insulated on one face and subjected to convective and radiative cooling at the other face. The plane slab has a uniform initial temperature and the ambient environment as well as the fluid temperatures are assumed to be constant. Heat conduction is considered to be one dimensional. Results are presented in dimensionless charts over a wide range of parameters.

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Instationäre Wärmeleitung mit gleichförmiger Wärmeerzeugung in einer Platte bei Kühlung durch Konvektion und Abstrahlung

Zusammenfassung Zur Ermittlung der instationären Temperaturverteilung in einer Platte mit gleichförmiger Wärmeerzeugung wird ein Differenzverfahren vierter Ordnung nach Runge-Kutta angewendet. Die ebene Platte ist einseitig isoliert und wird auf der anderen Seite durch Konvektion und Abstrahlung gekühlt. Zu Beginn befindet sich die Platte gleichförmig auf einer bestimmten Anfangstemperatur, die Temperaturen der umgebenden Objekte sowie des Fluids sind ebenfalls konstant. Der Wärmeleitungsvorgang sei eindimensional. Die Ergebnisse sind in dimensionsloser Form für einen weiten Parameterbereich in Diagrammform dargestellt.

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Nomenclature Bi Biot number - c specific heat - h heat transfer coefficient - k thermal conductivity - L half thickness of plate - Nr Radiation number - T absolute temperature - t time - F _o dimensionless time - T _e environment temperature - T _f fluid temperature - T _i initial plate temperature - u internal energy generation number - u internal energy generation - x space coordinate - x* dimensionless space coordinate Greek symbols thermal diffusivity - emissivity - dimensionless temperature - density - Stephan-Boltzman constant     

Approximate analysis for transient heat conduction with radiation in a slab

In an attempt to minimize the numerical computations associated with the solution of transient heat conduction with radiation in a slab, a perturbation type of analysis is being applied to the temperature field and radiation heat flux simultaneously. The resulting partial differential equations for the perturbation functions for the temperature are solved in explicit forms by use of the energy integral methods, while the radiation heat flux is determined by an appropriate scheme of approximating the temperature distribution in the slab. Included in the analysis are the effects of the parameters: the optical thickness, the ratio of conduction transport to radiation and the wall emissivity. It is found that, in a wide range of these governing parameters, the results compare very favorably with those obtained by the numerical solution of the formulated integro-differential equation. With the present analysis, the temperature, conduction and radiation heat fluxes can be predicted without resorting to lengthy numerical analysis.     

Interaction of heat transfer by conduction and radiation in a nongray planar medium

This paper considers the problem of the steady energy transfer due to combined effects of conduction and radiation in a medium with frequency dependent properties. The particular problem studied is the one-dimensional energy transfer in an absorbing, emitting, and conducting medium capable of generating heat which is bounded by two black plates. The analysis is restricted to an absorption coefficient x_v of theMilne-Eddington type, i.e., x_v(T)=αv β(T), and the frequency dependence of α(v) is approximated by a rectangular model. Temperature distribution and heat fluxes are reported in the paper for two models of spectral absorption coefficient and are compared with those for the gray case.     

A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media

Summary A new meshless method is developed to analyze steady-state heat conduction problems with arbitrarily spatially varying thermal conductivity in isotropic and anisotropic materials. The analog equation is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator can be employed to take the place of the fundamental solutions related to the original governing equation. Next, the particular solution is approximated by using radial basis functions, and the corresponding homogeneous solution is solved by means of the virtual boundary collocation method. As a result, a new method fully independent of mesh is developed. Finally, several numerical examples are implemented to demonstrate the efficiency and accuracy of the proposed method. The numerical results show good agreement with the actual results.This work was supported by the National Natural Science Foundation of China (No. 10472082) and Australian Research Council.     

A test fixture for mapping thermal conduction in composites under transient and steady-state conditions

A test fixture has been designed that allows the measurement of steady-state and transient temperature profiles in composites or other types of anisotropic or orthotropic materials under pure conduction in a vacuum. The uniqueness of this approach is the ability to study heat source to composite interface techniques by comparison of surface temperature contours and temperature versus time profiles. Localized changes in these conditions can be evaluated and the most successful technique can be adapted to larger parts, i.e., printed wiring boards (PWBs), electronic enclosure thermal walls, etc.     

Heat transfer for laminar flow in internally finned pipes with different fin heights and uniform wall temperature

An analysis is presented for fully developed laminar convective heat transfer in a pipe provided with internal longitudinal fins, and with uniform outside wall temperature. The fins are arranged in two groups of different heights. The governing equations have been solved numerically to obtain the velocity and temperature distributions. The results obtained for different pipe-fins geometries show that the fin heights affect greatly flow and heat transfer characteristics. Reducing the height of one fin group decreases the friction coefficient significantly. At the same time Nusselt number decreases inappreciably so that such reduction is justified. Thus, the use of different fin heights in internally finned pipes enables the enhancement of heat transfer at reasonably low friction coefficient.Nomenclature A_f dimensionless flow area of the finned pipe, Eq. (8) - a_f flow area of the finned pipe - C_p specific heat at constant pressure - f coefficient of friction, Eq. (12) - H₁, H₂ dimensionless fin height h₁/r_o h₂/r_o - h₁, h₂ fin heights - average heat transfer coefficient at solid-fluid interface - KR fin conductance parameter, k_s/k_f - k_f thermal conductivity of fluid - k_s thermal conductivity of fin - l pipe length - mass flow rate - N number of fins - Nu Nusselt number, Eqs. (15) and (16) - P pressure - Q total heat transfer rate at solid fluid interface - Q_f1, Q_f2 heat transfer rate at fin surface - q_w average heat flux at pipe-wall, Q/(2 r_ol) - R dimensionless radial coordinate r/r_o - Re Reynolds Number, Eq. (13) - r radial coordinate - r_o radius of pipe - r₁, r₂ radii of fin tips - T temperature - T_b bulk temperature - U dimensionless velocity, Eq. (2) - U_b dimensionless bulk velocity - u_z axial velocity - z axial coordinate - angle between the flanks of two adjacent fins - half the angle subtended by a fin - angle between the center-lines of two adjacent fins - angular coordinate - dynamic viscosity - density - dimensionless temperature, Eq. (6) - _b dimensionless bulk temperature     

A numerical and experimental investigation of the interactions between a non‐uniform planar array of incompressible free jets

The interaction between multiple incompressible air jets has been studied numerically and experimentally. The numerical predictions have been first validated using experimental data for a single jet configuration. The spreading features of five unequal jets in the configuration of one larger central jet surrounded by four smaller equi‐distant jets, have been studied, for different lateral spacing ratios of 1.5, 2.0 and 2.5 and a central jet Reynolds number of 1.24×105 (corresponding to a Mach number of 0.16). Flow of five equal jets has also been simulated, for the sake of comparison. The jet interactions commence at an axial distance of about 3–4 diameters and complete by an axial distance of about 10 diameters for the lowest spacing ratio of 1.5. For larger spacing ratios, the length required for the start and completion of jet interaction increase. Peripheral jets bend more towards the central jet and merge at a smaller distance, when their sizes are smaller than that of the central jet. The entrainment ratio for multiple jets is higher than that for a single jet. Excellent agreement is observed between the experimental data and theoretical predictions for both mean flow field and turbulent quantities, at regions away from the jet inlet. The potential core length and initial jet development, however, are not predicted very accurately due to differences in the assumed and actual velocity profiles at the jet inlet. Copyright © 2004 John Wiley & Sons, Ltd.     

Flow-rate limitation in a uniform thin-walled collapsible tube,with comparison to a uniform thick-walled tube and a tube of tapering thickness

Previous experiments on a tapered-thickness tube showed qualitatively different behaviour from that exhibited by a uniform thick-walled tube. To understand whether the taper or the thinner wall was responsible, similar aqueous flow-limitation experiments were conducted on a uniform thin-walled tube of the same material, with all other experimental set-up the same. As in the thick tube, there was a dramatic reduction in flow-rate when collapse and flow limitation started, but during external pressure reduction, the limited flow-rate progressively increased, so that as in the tapered-thickness tube, there was little flow-rate increase when collapse ceased. Hysteresis was thus a prominent feature of the relationship between flow-rate and pressure drop along curves of constant upstream transmural pressure. Flow-rate limitation was mainly accompanied by large-amplitude self-excited oscillation for both increasing and decreasing external pressure, to an even greater extent than in the tapered-thickness tube. Clusters of points sharing the same pair of upstream transmural pressure and upstream driving pressure values were found, indirectly implying as in the tapered-thickness tube that the flow-limited flow-rate for a given pressure drop was not uniquely determined by upstream transmural pressure. Negative effort dependence was observed in all three tubes, but in the thin tube, as in the tapered-thickness tube, it was obscured for some values of upstream transmural pressure where low-frequency single-collapse-per-cycle oscillations occurred. Thus, the qualitatively unique properties of the tapered-thickness tube appear to be confined to the relative lack of hysteresis, and the oscillatory regime in which collapse ceased before the downstream end. The rest of the observed behaviours seem to be characteristic simply of more compliant tubing.     

Heat transfer in a hydromagnetic flow between two porous disks—One rotating and other at rest,under uniform suction

Heat transfer in the flow of a conducting Fluid between two non-conducting porous disks (—one is rotating and other is stationary) in the presence of a transverse uniform magnetic field and under uniform suction, is studied. Asymptotic solutions are obtained for R«M ². The rate of Heat flux from the disks and the temperature distribution are investigated. It is observed that the temperature distribution and heat flux increase with the increase of magnetic field.     

Interaction between the near wake and the cross-section variation of a circular cylinder in uniform flow

The flow around a circular cylinder with a cross-section variation is experimentally investigated. Particle Image Velocimetry (PIV) is used to scrutinize the interaction of the cylinder’s wall with its near wake. The Reynolds number based on the cylinder’s diameter and freestream velocity is 80 × 10³, corresponding to the upper subcritical flow regime. At a forcing Strouhal number of St _f = 0.02, the maximum vorticity level around the cylinder is reduced by more than 50% as compared to its uncontrolled value. The topology of the bulk flow confined between the primary vortical structure and the cylinder surface is modified resulting in substantial drag reduction.     

Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid

We study the modelling of purely conductive heat transfer between a porous medium and an external fluid within the framework of the volume averaging method. When the temperature field for such a system is classically determined by coupling the macroscopic heat conduction equation in the porous medium domain to the heat conduction equation in the external fluid domain, it is shown that the phase average temperature cannot be predicted without a generally negligible error due to the fact that the boundary conditions at the interface between the two media are specified at the macroscopic level.Afterwards, it is presented an alternative modelling by means of a single equation involving an effective thermal conductivity which is a function of point inside the interfacial region.The theoretical results are illustrated by means of some numerical simulations for a model porous medium. In particular, temperature fields at the microscopic level are presented.Roman Letters _sf interfacial area of thes-f interface contained within the macroscopic system m² - A _sf interfacial area of thes-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - g vector that maps to _s , m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l _s,l _f microscopic characteristic length m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for thef-phase at thef-s interface - n outwardly directed unit normal vector at the dividing surface. - R ₀ REV characteristic length, m - T _i macroscopic temperature at the interface, K - error on the external fluid temperature due to the macroscopic boundary condition, K - T ^* macroscopic temperature field obtained by solving the macroscopic Equation (3), K - V averaging volume, m³ - V _s,V _f volume of the considered phase within the averaging volume, m³. - _mp volume of the porous medium domain, m³ - _ex volume of the external fluid domain, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² - x, z spatial coordinates Greek Letters _s, _f volume fraction - ratio of the effective thermal conductivity to the external fluid thermal conductivity - ^* macroscopic thermal conductivity (single equation model) kcal/m s K - _s, _f microscopic thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding toT ^*, K - spatial deviation temperature K - error in the temperature due to the macroscopic boundary conditions, K - ^* _i macroscopic temperature at the interface given by the single equation model, K - spatial average - ^s , ^f intrinsic phase average.     

Proportional and derivative control for steady-state vibration mitigation in a piecewise linear beam system

A control using Proportional and/or Derivative feedback (PD-control) is applied on a piecewise linear beam system with a flushing one-sided spring element for steady-state vibration amplitude mitigation. Two control objectives are formulated: (1) minimize the transversal vibration amplitude of the midpoint of the beam at the frequency where the first harmonic resonance occurs, (2) achieve this in a larger (low) excitation frequency range, where the lowest nonlinear normal mode dominates the response. Experimentally realizable combinations of PD-control are evaluated for both control objectives. Eventually objective (1) is realized by applying proportional control only, whereas derivative control is selected to realize objective (2). The vibration reduction that is achieved in simulations and validated by experiments is very significant for both objectives. Current results obtained with active PD-control are compared with earlier results obtained using a passive dynamic vibration absorber.     

On a supplementary conservation law for the governing equations in a rigid,magnetizable body with thermal and electrical conduction

Summary We consider a rigid heat conductor at rest immersed in an electromagnetic field. The interactions between thermodynamic and electromagnetic fields are decribed, in the frame work of the theory proposed by I. Müller [1]. Some properties related to the model equations proposed are pointed out. Specifically we are able to show as the governing equations may be written in symmetric and conservative form so that the Cauchy problem results well posed.

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Sommario Si considera un corpo rigido in quiete, conduttore di calore, immerso in un campo elettromagnetico esterno. Nell'ambito della teoria termodinamica proposta da I. Müller [1] si mettono in evidenza alcune proprietà del modello proposto che può essere scritto sotto forma simmetrica e conservativa assicurando, così, la buona posizione del problema di Cauchy.

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Research partially supported by C.N.R. through the G.N.F.M.     

Finite-part integral and boundary element method for interaction between two parallel planar cracks in three-dimensional finite bodies

By using the Somigliana representation and the concepts of finite-part integrals, a set of hypersingular integral equations of the interaction between two parallel planar cracks in a three-dimensional finite body subjected to arbitrary loads is derived, and then its numerical method is proposed by the finite-part integral method combined with the boundary element method. According to the analytic theory of hypersingular integral equations, the square root models of displacement discontinuities in the elements near the crack front are applied, and thus the computational precision is raised. Based on this, the stress intensity factors can be directly calculated. Finally, the stress intensity factors of several typical interaction problems are calculated.     

Reflection and transmission of a transient,elastic, line-source excited SH wave by a planar,elastic bonding surface in a solid

Closed-form time-domain expressions are obtained for the particle displacement of the elastic wave motion generated by a two-dimensional SH-wave line source and reflected and transmitted by a planar, elastic bonding interface of two homogeneous, isotropic, semi-infinite, perfectly elastic solids. The properties of the elastic bonding interface are characterized by a matrix of ‘spring coefficients’ through which the traction on each of the two faces is linearly related to the particle displacement of either of the two faces. The solution is constructed with the aid of (an extension of) the modified Cagniard method. The obtained solution of the forward model is believed to be of importance to the inverse problem that aims at reconstructing the elements of the matrix of ‘spring coefficients’ from measured values of the reflected and/or the transmitted wavefield quantities at a number of positions.     

Development of a distributed dislocation dipole technique for the analysis of multiple straight,kinked and branched cracks in an elastic half-plane

A distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half plane has been developed. The dipole density distribution is represented with a weighted Jacobi polynomial expansion where the weight function captures the asymptotic behaviour at each end of the crack. To allow for opening and sliding at crack kinking and branching the dipole density representation contains conditional extra terms which fulfills the asymptotic behaviour at each endpoint. Several test cases involving straight, kinked and branched cracks have been analysed, and the results suggest that the accuracy of the method is within 1% provided that Jacobi polynomial expansions up to at least the sixth order are used. Adopting even higher order Jacobi polynomials yields improved accuracy. The method is compared to a simplified procedure suggested in the literature where stress singularities associated with corners at kinking or branching are neglected in the representation for the dipole density distribution. The comparison suggests that both procedures work, but that the current procedure is superior, in as much as the same accuracy is reached using substantially lower order polynomial expansions.