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... • Time delay phenomena occurring in the interconnections of different parts of a system: As for such phenomena, authors in [6] used a space-averaging technique and the method of characteristics to propose a time-delay system modelling the flow temperatures of a heat exchanger; they believe that "time delay phenomena naturally occur in the interconnections of different parts of a system, as propagation of matter is not instantaneous. In particular, it occurs in tubular heat exchangers, which are very common devices in industry". ...

This paper focus on the heat recovery from the metallurgical and mining wastes. We propose and study a new and more realistic mathematical model for heat recovery from molten slag. Our model is based on time delay differential equations. In the theoretical part, we prove that a unique solution exists to the mathematical problem. In the numerical part, we establish an algorithm based on explicit fourth order Runge-Kutta method with delay; the new feature is that the delay must be larger enough than the step of integration. Compared to the classical model (without time delay), the numerical test proves that our model is more efficient and industrially more profitable.

... By utilising the Laplace transformation method, a dynamic analysis of a PHE is presented in [32] . In [33] , a heat exchanger model as a coupled first order hyperbolic PDEs is reformulated as time delay equations by using the method of characteristics. Moreover, in [34] , an accurate low-dimensional mathematical model of a compact heat exchanger based on the finite volume approach is derived, where a new interpolation scheme based on an approximate steady-state solution of the PDEs is introduced. ...

Understanding the dynamic properties of a plate heat exchanger plays an essential role when developing smart control algorithms for modern energy-efficient district heating and cooling applications. In this work, the dynamic behaviour of a counterflow plate heat exchanger is studied in order to understand its transient responses due to inlet temperature disturbances at different fluid flow configurations. For this purpose, a suitable theoretical model has been proposed. The temperature transients are evaluated numerically by solving a lumped-parameter system representing the derived 1D model of the studied plate heat exchanger. The numerical results revealed that different fluid flow configurations considerably influence the transient temperature responses as well as the overall temperature drop and heat transfer rate when following the cold down process of a fluid travelling from the inlet to the outlet of the plate heat exchanger hot fluid side. The predicted transients are experimentally verified by conducting a series of systematic tests using infrared thermography technology. The analysis shows that the proposed model is in good agreement with the experiments. The results of the thermal imaging measurements also provide a more in-depth insight into the temperature distribution and its transient front propagation along the fluid flow channels. In future studies, such an experimental approach is valuable to identify the critical temperature zones when predicting the thermal fatigue life-time of brazed plate heat exchangers and as well to extend the proposed theoretical model to include the 2D local heat transfer coefficient distribution obtained by the IR thermography.

How to create proper digital twin of plate heat exchange. The chapter will investigate alternatives for modeling and propose the best choices in modeling to trade-off between accuracy of the calculation vs. speed. Lump sum vs. distributed model, how to make plate model discrete, temperature distribution models, and details in heat exchange wall modeling will be considered. Last part would focus on experimental evaluation of the simulation using IR temperature vision measurements. Conclusions would take concrete examples of dynamic responses of heat exchange units, comparing numeric simulation with experimental results.

The heat exchangers are frequently used as constructive elements in various plants and their dynamics is very important. Their operation is usually controlled by manipulating inlet fluid temperatures or mass flow rates. On the basis of the accepted and critically clarified assumptions, a linearized mathematical model of the cross-flow heat exchanger has been derived, taking into account the wall dynamics. The model is based on the fundamental law of energy conservation, covers all heat accumulation storages in the process, and leads to the set of partial differential equations (PDE), which solution is not possible in closed form. In order to overcome this problem the approach based on physical discretization was applied with associated time delay on the positions where it was necessary and unavoidable. This is quite new approach, which represent the further extension of previous results which did not include significant time delay existing in the working media. Simulation results, were derived, showing progress in building such a model suitable for further treatment from the position of analysis as well as the needs for control synthesis problem.

This paper establishes the equivalence between systems described by a single
first-order hyperbolic partial differential equation and systems described by
integral delay equations. System-theoretic results are provided for both
classes of systems (among them converse Lyapunov results). The proposed
framework can allow the study of discontinuous solutions for nonlinear systems
described by a single first-order hyperbolic partial differential equation
under the effect of measurable inputs acting on the boundary and/or on the
differential equation. An illustrative example shows that the conversion of a
system described by a single first-order hyperbolic partial differential
equation to an integral delay system can simplify considerably the solution of
the corresponding robust feedback stabilization problem.

The operation of heat exchangers and other thermal equipment in the face of variable loads is usually controlled by manipulating inlet fluid temperatures or mass flow rates. This is a fundamental study of the basic issues regarding state and output controllability in such systems. A numerical method based on finite-differences is developed to approximate infinite-dimensional equations by finite-dimensional ones for the study of a conduction–convection system. The dynamics of a single-pass cross-flow heat exchanger with simultaneous advection, convection and conduction, in which water and air are the in- and over-tube fluids, respectively, is represented by a coupled set of partial differential equations. The numerical method is used to analyze the behavior of the heat exchanger equations. Using the water or air inlet temperature as the manipulated variable leads to a linear problem, and for the water flow rate it is non-linear. Controllability results for different choices of the manipulated variable are presented.

In this paper, the problem of remote output stabilization of networked control systems is investigated. The network is considered as a time-varying delay in the communication channel. An average model of the delay dynamics is supposed to be known and the unpredicted events occurring on the network are introduced as a random input in these dynamics. We propose a constructive control scheme where the deterministic aspect of the network is explicitly taken into account in a predictor-based feedback law. A stochastic descent algorithm is then introduced to set the controller gain according to the non-deterministic part of the delay dynamics. Some simulation results are also presented

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 counter-flow 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.

Fluid networks
are characterized by complex interconnected flows, involving high order nonlinear dynamics and transport phenomena.
Classical lumped models
typically capture the interconnections and nonlinear effects but ignore the transport phenomena, which may strongly affect the transient response. To control such flows with regulators of reduced complexity, we improve a classical lumped model (obtained by combining Kirchhoff’s laws and graph theory) by introducing the effect of advection as a time delay. The model is based on the isothermal Euler equations to describe the dynamics of the fluid through the pipe.
The resulting hyperbolic system of partial differential equations (PDEs) is diagonalized using
Riemann invariants to find a solution in terms of delayed equations, obtained analytically using the method of the characteristics. Conservation principles are applied at the nodes of the network to describe the dynamics as a set of (possibly non linear) delay differential equations.
Both linearized and nonlinear Euler equations are considered.

The author rejects, momentarily, actual experimentation with real heat exchangers for logical cogitation and arm-chair experiments. Observed as mental images or models, these heat exchangers reveal their basic characteristics—characteristics that will be partially obscured in an actual experiment by secondary effects inherent in the equipment. These basic characteristics, once perceived, may be used to correlate by means of frequency-response diagrams many results heretofore diverse and seemingly unrelated, that have been reported in the literature. Included among such results are not only the dynamics of various types of heat exchangers but also the thermal dynamic interaction between fluids and the confining pipe used for their transport. In addition, the author warns that all too frequently the use of the average temperature in heat-exchanger dynamics is based on faulty logic.

In this paper, a nonlinear observer of the thermal loads applied to the helium bath of a cryogenic refrigerator is proposed. The thermal loads represent a time-varying thermal disturbance expected to take place in future tokamaks refrigerators such as those used in the cooling systems for the International Thermonuclear Experimental Reactor (ITER) or the Japan Torus-60 Super Advanced (JT-60SA). The proposed observer can serve as a monitoring tool for cryogenic operators and/or in observer-based advanced control strategies. The observer is based on a part of the nonlinear model of the refrigerator. The paper details how the physical model of the Joule–Thompson cycle is obtained and the structure of the observer and validates its performance using experimental data.

The exit fluid temperature responses are presented for a unit step increase in the entrance temperature of either of the fluids of a counterflow heat exchanger. The exit temperature response histories are functions of four parameters, three of which are commonly used to define the steady-state temperature distributions in the exchanger. The responses are found using a finite difference method and are represented by simple empirical equations for a range of the four parameters believed appropriate for many technical applications.

The analysis of the transient behavior of shell and tube heat exchangers has been carried out by many researchers. Compared with the finite-difference procedure, the method presented in this note can be used to calculate directly the transient responses at arbitrary values of z and x, without performing many-step calculations pertinent to the selected time and space steps. Being different from other analytical procedures that can only deal with the transient responses to a step input change, this semi-analytical method can be applied to predicting the transient behavior subject to arbitrary inlet temperatures is shell and tube heat exchangers with parallel or countercurrent flow.

Control of a tubular heat exchanger

- J O M Bakosova
- M Kacur