Equipartition of forces as a lower bound on the entropy production in heat exchange
Introduction
Bejan [3] has discussed the design of a wide variety of heat exchangers that operate with minimum entropy production or maximum second law efficiency. This work is also concerned with heat exchange at minimum entropy production. We shall not, however, find the entropy production for given boundary conditions, as Bejan [3] does. We shall determine (ideal) boundary conditions that are compatible with a state of minimum entropy production for a given duty.
Our problem must not be confused with the typical industrial problem; that is to find the minimum area of heat exchange by varying the flow rate and the outlet temperature of the coolant. This question has its direct answer by solving the energy balance. We shall instead first find ideal boundary conditions from the second law of thermodynamics. The energy balance is next applied in order to realize the ideal conditions. It is of interest to study the ideal heat exchange process, to see how far it is possible to increase the second law efficiency of heat exchange.
The problem we raise, was solved on general grounds by Sauar et al. [14]. Minimum entropy production for a process with a given production (duty), was obtained with a constant driving force. The force was defined by irreversible thermodynamics. A constant driving force gives a lower bound on the process, and enables us to measure the distance of any real process to the most efficient one. No restriction was found for the transport coefficient [14] provided that the optimization problem was formulated according to the rules of irreversible thermodynamics. The constant force represents the boundary conditions that we are looking for. The proof has not yet been applied to heat exchangers, which is of interest here.
Some questions related to the proof arise from the literature. Tondeur and Kvaalen [18], Tondeur [17] stated that the local entropy production rate is constant for heat exchange. Their force for heat transfer was Δ(1/T). With a constant heat transfer coefficient, this is equivalent to a constant force. It has been assumed that the thermal conductivity must be constant in order for the optimal driving force to be constant [17]. Haug-Warberg [6] has recently argued the same. On the other hand, Bejan [1], [2], and Minta and Smith [9] in their construction of a helium liquefaction cycle found that an optimal cycle was obtained when the temperature difference between the media, over the average temperature, ΔT/T, was everywhere constant. In a thermal design study of LNG heat exchangers, Fredheim [5] found that the exergy loss in the heat exchanger was minimum when the temperature difference between the heating and the cooling medium was constant. Minimum exergy loss is equivalent to minimum entropy production. The first question that arises from the literature is therefore: If a constant driving force is characteristic for a maximum efficiency, how is this force defined? We shall furthermore see in more detail that the coefficient need not be constant for the constant force criterion to be true, and that the force is defined completely within the framework of irreversible thermodynamics once a choice is made for the flux.
Section snippets
The system
The system consists of a simple heat exchanger with two fluids separated by a thin metal plate. A sketch of the unit and an enlarged control volume is given in Fig. 1. The heat exchanger might for instance cool a hydrocarbon oil with water. The fluids flow at constant rates in the z-direction and we assume plug flow with perfect thermal mixing in the x- and y-directions. The fluids can pass each other in co- or counter-current fashions. The heat flux is directed from one fluid to the other; in
Calculations
Four different cases of heat exchange were calculated, all at constant (mass flows), Thl and Thr. The subscripts c and h mean cold and hot fluid, while hl and hr refer to the left and right side of the hot fluid, respectively (see Fig. 1). The corresponding temperatures for the cold fluid were Tcl and Tcr.
The following numerical data were chosen to demonstrate typical effects. The duty of the heat exchanger was Q=60 kW, the heat transfer coefficient was , the mass flow of
The reference cases
The results are presented in Table 1. Consider first cases 1 and 2, the co-current and the counter-current modes of operation. We see that the area requirement is 19% less at counter-current than at co-current heat exchange for the same entropy production. Table 1 shows that the thermodynamic driving force (Δ(1/T)) through the heat exchanger is closer to constant in case 2 than in case 1, but it is not completely constant. At the inlet of the heat exchanger it is , at the outlet the
Conclusion
We have shown, using entropy production minimization, why it is more advantageous, from the point of the second law, to operate a heat exchanger in counter-current than in co-current mode. We have seen by analyzing a numerical example, that a family of operating lines called isoforce operating lines may be used to assess the efficiency of heat exchangers. Area reductions can be obtained by replacing normal operation by isoforce operation, but these savings must be traded off by frictional
Acknowledgements
Lars Nummedal is grateful for a grant given by The Norwegian Research Council. Dick Bedeaux is thanked for help in clarifying the mathematics.
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