Variational Approach to 2D and 3D Heat Conduction Modeling

The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.

Variational Approach to Heat Conduction Modeling

It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.

Experimental Investigation of Pyrolysis Process of Woody Biomass Mixture

This paper describes an experimental investigation of pyrolysis of woody biomass mixture. The mixture consists of oak, beech, fir, cherry, walnut and linden wood chips with equal mass fractions. During the experiment, the sample mass inside the reactor was 10 g with a particle diameter of 5-10 mm. The sample in the reactor was heated in the temperature range of 24-650oC. Average sample heating rates in the reactor were 21, 30 and 54 oC/min. The sample mass before, during and after pyrolysis was determined using a digital scale. Experimental results of the sample mass change indicate that the highest yield of pyrolytic gas was achieved at the temperature slightly above 650oC and ranged from 77 to 85%, while char yield ranged from 15 to 23%. Heating rate has significant influence on the pyrolytic gas and char yields. It was determined that higher pyrolysis temperatures and heating rates induce higher yields of pyrolytic gas, while the char mass reduces. Condensation of pyrolytic gas at the end of the pyrolysis process at 650oC produced 2.4-2.72 g of liquid phase. The results obtained represent a starting basis for determining material and heat balance of pyrolysis process as well as woody biomass pyrolysis equipment.