Solutions of nonlinear boundary value problem with applications to biomass thermal conversion

Abstract

Bioenergy is one of the widespread renewable energy sources. Energy from biomass thermal conversion can reduce greenhouse emissions and have a positive effect on climate change. Biomass conversion is generally carried out in reactors of cylindrical shape. From a modelling point of view many factors have to be taken into account in order to optimize thermal efficiency of the conversion process. One of the important methods for the analysis of complex fluid flows is hydrodynamic stability theory. Base flow solution in classical hydrodynamic stability problems is usually found as a simple analytical solution of the equations of motion. Biomass conversion problems lead to nonlinear boundary value problems, which can be either solved numerically or analyzed using the bifurcation theory. In the present paper we analyze a mathematical model of heat transfer in the presence of nonlinear heat sources. This model includes the study of positive solutions to a nonlinear boundary value problem with certain boundary conditions. The equations in a problem contain several parameters, which essentially affect the behaviour and the number of solutions. Bifurcation analysis of the problem, conducted with respect to the parameters, allows obtaining somewhat precise results on the number of positive solutions. Generally, two, one and zero positive solutions are possible, depending on the values of the parameters. The obtained solutions represent base flow for the hydrodynamic stability problem, which can be solved with the objective to identify the factors affecting the conversion process.

On a System Without Critical Points Arising in Heat Conductivity Theory

“…The end point of any trajectory corresponds to (x(1), y(1)). If the end point is on the axis x = 0, then this trajectory corresponds to a solution of the BVP (3), (5). Then equation ( 4) is…”

“…This equation together with the boundary (3) if F > F 0 , then the problem (20), (5) has no positive solutions.…”

“…An alternative method can be used to study bifurcations in the problem (20), (5). Consider the Cauchy problem…”

“…The problem is to detect the number of positive solutions to BVP and trace the change of this number under the influence of built-in parameters. For more details the interested reader can consult the papers [11], [5], [4]. In this paper we consider one of these problems.…”

“…A solution T (t) (T is the temperature, t is treated here as a spatial variable) is supposed to be C 2 -smooth function. For general reading about problems, arising in heat conductivity theory, one may consult [1], [2], [3], [5], [16].…”