Flow-based control of temperature in long ducts

Flow-based control of a thermal system with a long duct and heat loss to the environment is analyzed. A Proportional-Integral controller is used to regulate the duct outlet temperature by using the flow velocity as control input. The one-dimensional energy equation in Eulerian and Lagrangian forms are numerically solved. The non-linear dynamics can be represented by an integral equation in terms of the residence time which acts as a delay. A linear stability analysis leads to a characteristic transcendental equation which is examined for different orders of the residence time. Pontryagin’s theorem on the zeros of exponential polynomials is used to obtain stability maps as a function of system parameters. Numerical simulations are performed to verify the predictions, determine super- and sub-critical instabilities, and evaluate the amplitude and frequency of limit-cycle oscillations.