A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling

Abstract

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower temperature leaves the tank through another pipe. We propose a one-dimensional mathematical model that consists of a nonlinear PDE for the temperature along the solid body, coupled to a linear ODE for the temperature in the tank, the boundary and the initial conditions. All equations are converted into a dimensionless form reducing the input parameters to three dimensionless numbers and a dimensionless function. A steady-state analysis is performed. To solve the transient problem, a nontrivial numerical approach is proposed whereby the differential equations are first discretized in time. This reduces the problem to a sequence of nonlinear two-point boundary value problems (TPBVP) and a sequence of linear algebraic equations coupled to it. We show that knowing the temperature in the system at time level n − 1 allows us to decouple the TPBVP and the corresponding algebraic equation at time level n. Thus, starting from the initial conditions, the equations are decoupled and solved sequentially. The TPBVPs are solved by FDM with the Newtonian method.

Keywords

• nonlinear
• coupled PDE-ODE system
• heat equation
• TPBVP
• finite difference method