# Steady-state and Dynamic Response Simulation of a One-Dimensional Heated Rod using Orthogonal Collocation

### Abstract

This paper presents the steady-state and dynamic response simulation of a one-dimensional heat conduction system starting from the parabolic partial differential equation (PDE) modelling the system. The PDE modelling the system was lumped using the orthogonal collocation method which is a method of weighted residuals. Both the linear and nonlinear cases were considered. The simulation of the lumped linear dynamic system was carried out based on the computation of the state transition matrix and the corresponding integral. The simulation of the lumped nonlinear system was done by numerical integration using the Semi-Implicit Third-Order Runge-Kutta method. It was found that using only 3 to 4 collocation points produced excellent results (both steady-state and transient response), when compared with the analytical results for the linear case. For the linear system formulation, the approach can be used to develop a low-order lumped model suitable for use in control design based on, for example, the classical PID controller method. For linear variation of thermal conductivity with temperature, it is shown that there may be significant errors in the the system dynamic response between the linear and nonlinear formulations for the system over a wide temperature operating conditions.

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*Journal of Engineering Research*,

*24*(2), 40-58. Retrieved from http://jer.unilag.edu.ng/article/view/577