On applicability of the boundary condition of the first kind in problems of heat transfer in heat-releasing granular beds

The range of applicability of the boundary condition of the first kind at entry into a heat-releasing granular bed has been investigated. It has been shown that in the cases where the heat-transfer process can be described within the framework of a one-temperature (homogeneous) model, it is allowable to use this condition for RePr >> 1. When it is necessary to allow for the phase-temperature difference, the boundary condition of the first kind can be used for the parametric value \overset \smallfrownQ \overset {\smallfrown}{Q} ≤ 0.02. In the remaining regimes, the generalized Danckwerts condition allowing for the preheating of the heat-transfer agent should be used.     

External heat transfer in infiltrated granular beds. 1. Analysis of a theoretical model

A model is proposed for external heat transfer in infiltrated fixed granular beds. The model is analyzed to derive simple equations for calculation of the heat-transfer coefficient.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 53, No. 4, pp. 580–585, October, 1987.     

Heat transfer in granular beds in radiative heat supply

The basic regularities of stationary heat transfer throughout the space of an infiltrated granular bed in radiative heat supply in cocurrent-flow (solar collector 1) and counterflow (solar collector 2) regimes have been investigated within the framework of a two-temperature model. The boundary layer of the third kind for the skeleton of particles at exit from the bed has been formulated; this condition allows for the degree of turbulence of the heat-transfer-agent flow. A quasihomogeneity criterion making it possible to evaluate the thermal state of a two-phase system has been introduced. The approximation dependences for calculation of the active-portion length, the bed’s resistance, the solar-collector efficiency, and the average relative phase-temperature difference have been established.     

Establishment of boundary conditions for problems of conductive-convective heat transfer in disperse systems

As a result of analysis of various boundary conditions for the elliptic-type heat conduction equation which describes a stationary thermal state of a filtered granular bed, it has been established that the first-kind condition used earlier in literature for the inlet to the bed is incorrect and responsible for the disturbance of the overall heat balance of the system. The necessity of using the Danckwerts conditions, which reflect the specificity of the input of a heat carrier into the bed and output from it in the presence of longitudinal conductive heat transfer, is shown. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 3, pp. 66–71, May–June, 2007.     

Effect of boundary conditions on the heat transfer law for a turbulent boundary layer

The heat transfer at an impermeable plate has been experimentally investigated under various boundary conditions. The conservativeness of the heat transfer law St₀=f(Re _T ^** ) is demonstrated for a monotonic increase of temperature and heat flux along the surface.     

External heat transfer in infiltrated granular beds. Experimental study

Experimental data was obtained on external heat transfer in infiltrated granular beds of coarse particles. It was determined that calcualtions with the proposed model agree satisfactorily with the experimental results.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 53, No. 6, pp. 919–923, December, 1988.     

Electrical simulation of boundary conditions in nonsteady problems op heat and mass transfer

A method of modeling heat- and mass-transfer problems with boundary conditions of the second and third kinds is described. Simplified model circuits are presented.     

Hydrodynamics and external heat transfer in granular beds with rotating cylinders placed in them

The results are presented from an experimental investigation of the effect of rotation of a cylinder placed horizontally in fixed and fluidized granular media on the structure of the layer adjacent to the wall and on the intensity of external heat transfer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 52, No. 1, pp. 5–9, January, 1987.     

Additional boundary conditions in unsteady-state heat conduction problems

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).     

Validity of the boundary conditions defined for measurements of the heat transfer coefficient on the basis of the thermal strain rate in specimens

The velocity of the dilatometric method of measuring the heat transfer coefficient for cylindrical surfaces in a transverse stream of liquid is confirmed theoretically.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 4, pp. 645–647, April, 1972.     

Additional boundary conditions in nonstationary problems of heat conduction

The integral method of thermal balance, with determining the front of temperature disturbance and introducing additional boundary conditions, is used to obtain analytical solutions of heat conduction problems for plate, cylinder, and sphere at the boundary conditions of the third kind. The distribution of isotherms and the velocities of their motion are analyzed.     

A direct boundary element approach for unsteady non-linear heat transfer problems

Unsteady non-linear problems are usually solved based on the time marching scheme together with iterative methods. In this paper, we employed an example to illustrate a new direct boundary element technique, which can provide accurate solutions of unsteady non-linear problems without the need for iterations. The principle and validity of the technique is demonstrated by considering an unsteady non-linear hyperbolic heat transfer problem, together with a fully implicit difference scheme in the time domain. It illustrates that the proposed non-iterative boundary element approach is numerically rather efficient for unsteady non-linear problems. Thus, it provides an alternative to traditional iterative methods for unsteady non-linear problems.     

Comparison of boundary collocation methods for singular and non-singular axisymmetric heat transfer problems

This paper presents a computational study of some boundary collocation solution methods for the Laplace equation in cylindrical coordinates with axisymmetry. The methods compared are (i) the direct boundary element method (BEM), (ii) the method of fundamental solutions (MFS) with fixed sources and (iii) the Trefftz method. Relative accuracy of these methods are compared for two test problems. The first problem is a simple problem of heat transfer through a cylindrical rod which is a standard benchmark problem in this field. The second problem deals with heat transfer in silicon melt for Czochralski (CZ) process which involves a singularity in the boundary conditions at the corner of the crystal–melt interface. All the three methods indicated above are highly successful for the simple (first) problem with MFS and Trefftz being simpler to implement than the BEM. However, the Trefftz method was not effective for the second problem due to the boundary singularity and the MFS showed oscillations near the singularity point. Hence the use of higher order non-conforming elements with accurate Gauss–Kronrod integration schemes in the direct BEM method was investigated. It was found that the boundary singularity does not deteriorate the accuracy of the results if this improved numerical integration procedure is used in the direct BEM. Hence higher order elements with Gauss–Kronrod integration schemes can be used for the solution of many free interface problems encountered in crystal growth.     

Nonsteady-state heat transfer in a high-temperature granular bed

Specific features of heat transfer in a high-temperature granular bed are studied based on a two-region model which takes into account the existence of an elevated porosity area.Academic Scientific Complex A. V. Luikov Heat and Mass Transfer Institute of the Academy of Sciences of Belarus, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 64, No. 4, pp. 421–425, April, 1993.     

A numerical method for heat transfer problems using collocation and radial basis functions

Simple, mesh/grid free, numerical schemes for the solution of heat transfer problems are developed and validated. Unlike the mesh or grid-based methods, these schemes use well-distributed quasi-random collocation points and approximate the solution using radial basis functions. The schemes work in a similar fashion as finite differences but with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no extra difficulty. © 1998 John Wiley & Sons, Ltd.     

Experimental investigation of heat transfer in axisymmetric volumes for boundary conditions of the second kind

Free-convection heat transfer is investigated in spherical volumes and in a horizontal cylinder with a constant heat flux density at the surface. Relations are obtained for the temperature distribution and the rate of heat transfer at the wall — liquid interface.     

General boundary element method for non-linear heat transfer problems governed by hyperbolic heat conduction equation

The general Boundary Element Method (BEM) for strongly non-linear problems proposed by Liao (1995) is further applied to solve a two-dimensional unsteady non-linear heat transfer problem in the time domain, governed by the hyperbolic heat conduction equation (HHCE) with the temperature-dependent thermal conductivity coefficients which are different in the x and y directions. This paper confirms that the general BEM can be used to solve even those non-linear unsteady heat transfer problems whose governing equations do not contain any linear terms in spatial domain.