Conduction, heat numerical schemes

Let us consider a conservative numerical scheme [7,12]. The calculation cell has the form of a parallelepiped. Mesh nodes corresponding to the vertices of the design volume are marked with integers. The values of temperature, thermal conductivity coefficients, heat capacity belongs to... 

On the basis of the theory of numerical methods and mathematical modeling the problem of the calculation and forecast of the distribution of the temperature field in a two-phase nanocomposite environment is solved. The mathematical statement of the problem is formulated as the integral equation of thermal balance with a heat flux taken into account, which changes according to Fourier s law. Jumps of enthalpy and heat conductivity coefficient are considered. Various numerical schemes and methods are examined and the best one is selected - the method of control volume. Calculation of the dynamics of the temperature field in the nanostructure is hold using the software. 

The usual practice in numerical analysis of the heat conduction equation (52) is connected with three-layer schemes. The values y x)... 

The numerical solution of problem (1) by means of iteration schemes can be done using the alternating direction scheme for the heat conduction... 

The usual practice in numerical analysis of the heat conduction equation (52) is connected with three-layer schemes. The values y3 1(x), y3 (x) and y3+1(x) of a grid function on the three time layers tj L, f and tj+1 are aimed at constructing such schemes. 

Lotkin (L10) gives a scheme for numerical integration of the heat conduction equation in a finite ablating slab, using unequal subdivisions in both space and time variables. Near the melting surface it is advantageous to choose rather small integration steps. Stability characteristics of the method are established. 

This paper deals with thermal wave behavior during frmisient heat conduction in a film (solid plate) subjected to a laser heat source with various time characteristics from botii side surfaces. Emphasis is placed on the effect of the time characteristics of the laser heat source (constant, pulsed and periodic) on tiiermal wave propagation. Analytical solutions are obtained by memis of a numerical technique based on MacCormack s predictor-corrector scheme to solve the non-Fourier, hyperbolic heat conduction equation. 

Heat waves have been theoretically studied in a very thin film subjected to a laser heat source and a sudden symmetric temperature change at two side walls. The non-Fourier, hyperbolic heat conduction equation is solved using a numerical technique based on MacCormak s predictor-corrector scheme. Results have been obtained for ftie propagation process, magnitude and shape of thermal waves and the range of film ftiickness Mid duration time wiftiin which heat propagates as wave. 

Wood WL, Lewis RW (1975) A comparison of time marching schemes for the transient heat conduction equation. Int J Numer Methods Eng 9 679-689... 

Gray and Lee give an elegant review of the theory of thermal explosions. The first numerical solution of the nonlinear heat conduction equation with zero-order Arrhenius kinetics was obtained by Zinn and Mader. They did not use a finite difference scheme, and their results have been used as a standard for comparison with the faster and more general treatments like those in the TEPLO or SIN codes. 

The Reynolds equation and the energy equation in the interfacial film and the conduction equation in the seal rings are solved numerically by the finite difference technique. These equations are coupled by the heat exchange conditions on the boundaries of their domains. They are integrated by using the Crank Nicholson scheme. 

In Chap. 7, the investigation on combustion stabihty is extended to propane-fueled catalytic microreactors, using the catalytic and gas-phase chemical reaction schemes of propane combustion on platinum proposed and validated in Chap. 4. The steady hetero-Zhomogeneous combustion of lean propaneZair and methaneZair mixtures in a platinum-coated, catalytic plane channel-flow microreactor were investigated at pressures of 1 and 5 bar, channel heights of 1.0 and 0.3 mm, and wall thermal conductivities of 2 and 16 WZmK. Stability limits were assessed as a function of fuel type, inlet velocity, and imposed external heat losses. Parametric studies were performed with a full-eUiptic, two-dimensional numerical model employing detailed gas-phase (homogeneous) reaction schemes for both fuels. 

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