PDE Observer Design for Counter-Current Heat Flows in a Heat-Exchanger

Abstract

In this work, we consider the estimation of temperature profiles along the pipes of a plate heat exchanger. The transport phenomena through the heat exchanger are modeled by hyperbolic partial differential equations (PDE) of first order in time and space. The counterflow heat exchange implies that the system is comprised of rightward (where the hot fluid circulates) and leftward (cold fluid pipes) hyperbolic PDE. The heat exchanged between the pipes of hot and cold fluid induces a coupling between the rightward and leftward equations, which increases the difficulty of solving the PDE system. The estimation objective is addressed by the design of an observer using a PDE approach, which uses boundary measurements to estimate the distributed profiles. The convergence of the observation error is established using Lyapunov analysis. Simulation results illustrate the efficiency of our method using a simulator with time-varying parameters validated on experimental data.

Modeling and Estimation of Thermal Flows Based on Transport and Balance Equations

“…The boundary conditions on T m are not necessary since equation (9) does not contain spatial derivatives of T m . Likewise, in [7], a reduced model can be easily derived from (11) by neglecting the dynamics of the temperature in the pipe, i.e., assuming that such a temperature is fixed by the values of the temperatures of cold and hot fluids, i.e., imposing that the r.h.s. of (6) is null.…”

“…In case of the reduced model (16) (i.e., with A = A 2 and B = B 2 ), the stability conditions of Theorem 1 can be greatly simplified according to [7] by using…”

“…for all t ≥ 0 with ℓ 1 , ℓ 2 , μ 1 , μ 2 ∈ ℝ to be suitably chosen. More specifically, in [7] it is proposed to satisfy (23) by using an LMI-based discretized approach to get μ 1 , μ 2 and then select ℓ 1 , ℓ 2 such that…”

“…Based on such equations, first we will focus on a model with the dynamics of the temperatures of cold fluid, hot fluid, and metal pipe. In addition, a reduced model is considered with only the temperatures of cold and hot fluid as state variables in line with [7]. Since such temperatures cannot be determined by direct, distributed measurements, it is fundamental to estimate them by using a few numbers of measures.…”

Distributed feedback control of a fractional diffusion process

“…In that case, it is necessary to estimate the states by means of a state observer using some measurements. However, it was not done here as the observer design for a PDE [46][47][48][49][50][51] and even more for a FPDE is com-plex and would have deserved to be the subject of a totally different study. If measurements are noisy, the measurements should be first cleaned from the highfrequency noise by a low-pass filter, then an observer based on the FPDE should be used.…”