Heat exchange systems with minimal irreversibility

Highlights

• Conditions of minimum dissipation of heat exchange systems with variable heat capacity of flows.

• Distribution of the contact surfaces between the heat exchangers, corresponding to the minimum dissipation.

• The class of dependences of the heat flow on the contact temperatures is expanded.

• The conditions of thermodynamic equivalence of two – and multithreaded systems are obtained.

• An algorithm for the synthesis of systems from two-flow heat exchangers is proposed.

Abstract

The conditions for minimum dissipation and the dependence of the minimum possible entropy production on the heat load are obtained for heat exchange systems with a given heat load and total heat transfer coefficient. The case of variable heat capacity of streams is considered. Distributions of heat exchange coefficients and heat capacity rates of the streams for which this minimum of irreversibility can be achieved are found. The problem is solved for a wide class of heat transfer kinetic laws in which the heat flow is proportional to the contact surface. The possibility of evaporation and condensation of streams is investigated. The concept of thermodynamic equivalence of dual-stream and multi-stream systems is introduced and the conditions that lead to such equivalence are obtained. Using this equivalence for Newtonian kinetics, optimal synthesis conditions are obtained for dual-stream and multi-stream systems with a given total heat transfer coefficient and heat load. It is shown how the solution can be implemented in a system of dual-stream heat exchangers.

Introduction

The thermodynamic approach to the analysis of technological systems (see e.g. Shnip, 2018, Kakac, 2018) allows one to identify the boundary of the set of realizable systems and to find their maximum capabilities (heat and mass transfer systems, heat and refrigeration machines, separation systems, chemical reactors, etc.). Most of these bounds are based on the relations of thermodynamics of reversible processes (Carnot efficiency, reversible work of Gibbs separation) and they are very important as ultimate limits. However, they are quite optimistic in comparison with the capabilities of real systems. The reason for this is that these reversible models do not take into account the intensity of flows, the size of the contact surfaces, and other factors related to the finite dimensions of the devices and required speeds of operation.

Attempts to take these factors into account and obtain the characteristics of heat and refrigeration machines that are optimal in the class of irreversible processes have led to the intensive development of optimization thermodynamics or finite-time thermodynamics (Andresen, 1983, Berry et al., 1999). Thus, along with the ideal reversible Carnot heat engine, the maximum power (Curzon-Ahlborn) heat engine (Novikov, 1958, Curzon and Ahlborn, 1975), as well as a fixed power heat engine with maximum efficiency were considered (Rozonoer and Tsirlin, 1983a, Rozonoer and Tsirlin, 1983b, Rozonoer and Tsirlin, 1983c). In all these cases, we solved the problem of an ideal heat engine with natural restrictions on the heat exchange coefficients of the working medium with its heat sources and on the power of the machine. We emphasize that other restrictions can be taken into account as well such as the amount of friction between the piston of the heat engine and the cylinder surface, heat loss to the environment, etc. (see e.g. Andresen and Huleihil, 2006).

For some processes, reversible bounds only make sense as unrealistic idealizations. One of the most important examples of such systems is heat exchange systems. The assessment of thermodynamic perfection (how close they are to reversible performance) of these is impossible without taking into account the limited contact surface (the total coefficient of heat exchange) and the required heat load (the amount of heat transferred per unit of time from hot to cold streams). To assess the perfection of such systems, an exergetic approach is used (see e.g. Brodjanskiy et al., 1988, Bosnjakovic, 1965), comparing the systems by the loss of exergy in each of them.

Exergy losses are proportional to the entropy production in the system and the ambient temperature T₀. In the exergetic analysis of a heat exchange system, the entropy production in the system is calculated using the known heat capacities of the streams and the temperatures of the streams at the inputs and outputs of the system without going into detail about how to construct a mechanism to do this, i.e. without solving the problem of system synthesis. The synthesis problem is defined as follows: Which conditions must be met by the heat exchange system so that the production of entropy in it (and hence the loss of exergy) under the given restrictions is minimal. This paper is devoted to solving this problem.

A significant number of studies have been devoted to the conditions for minimum dissipation in thermodynamic processes. Thus, in their celebrated work (Tondeur and Kvaalen, 1987) Tondeur and Kvaalen propose constancy of the rate of entropy production (“equipartition of entropy production”) as a condition for achieving minimal irreversibility for a certain class of systems, primarily with linear rate dependence. While this is a wonderfully simple conclusion which holds in many practical situations and is a good approximation in many others, it is frequently used indiscriminately also outside its region of applicability. In Tsirlin et al. (1998) derive conditions for minimum dissipation for arbitrary kinetics. And finally in Tsirlin et al. (2003) it is proved that minimum dissipation corresponds to constancy of the rate of entropy production only under certain conditions, and a class of kinetic regularities is distinguished for which this condition is valid. To use the results of these studies in our problem it was necessary to take into account the restrictions on the temperatures of all parts of the streams, changes in their heat capacities from temperatures, etc.

Another celebrated approach to minimizing dissipation in multi-stream heat exchange systems is pinch analysis (Linnhoff and Hindmarsh, 1983) (and possibly textbooks like Smith, 2005, Kemp, 2006). The method is based on aggregated heat loads of the entire system at individual temperatures arranged in (T, Q) plots analogous to e.g. our Fig. 3. As a result of this construction, the point where the temperature difference between heat supply and heat demand is minimal (the pinch point) is determined, and the desired structure of the system may de be concluded. This is typically obtained with given heat loads and given constant heat conductances for each element of the system. By contrast, our derivation below has at its disposal a total amount of heat conductance which may be distributed among the heat exchange units for total optimality. Further, we consider a wide range of heat conductance dependencies as well as include the possibility of phase change (boiling and/or condensation). It follows from relations derived below that the vicinity of the pinch point makes the maximum contribution both to the production of entropy and to the value of the required heat transfer coefficient. Thus the relations derived here can be viewed as the thermodynamic justification for pinch analysis.

This paper consists of two sections. The first is devoted to obtaining conditions for minimum dissipation in dual-stream systems for various forms of heat transfer dependencies and for fixed streams with variable heat capacity and with a given temperature change profile. These results are used in the second section where multi-stream systems are considered and conditions for thermodynamic equivalence of dual-stream and multi-stream systems are obtained. This equivalence makes it possible to implement multi-stream systems with minimal irreversibility using the structure of dual-stream heat exchangers.