# Minimizing entropy in thermal systems.

Minimizing Entropy in Thermal Systems

One of the most central teachings of classical engineering thermodynamics is the idea that the efficiency of a machine cannot exceed a certain theoretical limit. We know, for example, that the energy-conversion efficiency of a thermal power plant (Figure 1) cannot exceed the Carnot efficiency (1 - T.sub.0./T.sub.H.). Similarly, the coefficient of performance (COP) of a refrigeration plant cannot be greater than the COP of the Carnot refrigerator sandwiched between the same temperature extremes (1 - T.sub.L./T.sub.0.) - 1.

These theoretical limits of operation correspond to machines that operate reversibly. Of great practical importance is the discrepancy between the figure of merit of an actual machine and the ceiling value of the same figure of merit in the limit of reversible operation. The penalty for this discrepancy is most visible in a refrigeration plant, where the actual work input required per unit of refrigeration load is greater than the work input required by the corresponding Carnot refrigerator. The difference between the actual work and the Carnot work, that is, the penalty, represents the lost work, or in more current terminology, the irreversibility.

A theorem of classical thermodynamics that holds for any system in communication with the ambient at temperature T.sub.0 is the proportionality between the irreversibility and the entroply generated by the system: Irreversibility = T.sub.0.S.sub.gen

This is the Gouy-Stodola theorem, in memory of the French physicist G. Gouy and the Swiss engineer A. Stodola, who drew attention to it in 1889 and 1905, respectively.

Students of the history of engineering know that many great ideas were overlooked because of inertia in the established point of view. The Gouy-Stodola theorem is one idea that only now, almost a century later, is beginning to receive the attention it deserves. The proportionality between the irreversibility and S.sub.gen raises the status of the concept of entropy generation to that of a crucial parameter in the art of thermal design. For if we are serious about constructing efficient energy systems and conserving energy, we have no choice but to design for less and less entropy. Of course, the avoidance of entropy is only one of the problems in the designer's mind. The more central concern is connected to the economic aspects of the design. This connection is pursued in the field of thermoeconomics (see pp. 84-86).

The immediate design questions that follow from the Gouy-Stodola theorem are:

* What is all the entropy that is being generated by the system?

* How is S.sub.gen distributed over the system, or what components and subcomponents are most responsible for the entropy generated by the system?

* How can we redesign the most faulty (least reversible) components so that they continue to perform their system functions while generating less entropy?

These questions are pursued vigorously in the research area covered by the ASME Advanced Energy Systems Division. An important conclusion is that the overall entropy generation is| the sum of contributions where i = 1,2,...,n indicates all the components and subcomponents of the overall system. Figure 1 shows that if the system is a power or refrigeration plant, then the S.sub.gen value of the plant is the sum of the individual S.sub.gen values of all the components (e.g., heat exchangers, pumps, turbines, valves, etc.). At the next, more basic level, the S.sub.gen value of one such component--a heat exchanger--is due to the irreversibility contributed by all its tubes, baffles, etc. Finally, at an even more basic level, the S.sub.gen value of a single tube may be the aggregate effect of the entropy generated by each fin or rib attached to the tube surface.

Competing Irreversibilities

A solid understanding of the mechanism of entropy generation of each of the system compartments is essential in mapping out a strategy for decreasing the entropy generated by the overall system.

The entropy generated by a single heat exchanger tube, for example, is due to two effects: the transfer of heat along a finite temperature gradient across the tube wall; and the flow with friction, that is, the flow in the presence of a finite pressure drop along the tube. The most striking aspect of this mechanism is that these two effects compete against one another. A certain design change (say, a reductionin tube diameter) leads to a decrease in the heat-transfer irreversibility and an increase in the fluid-flow irreversibility.

There is an important trade-off between the two effects, that is, an optimum balance when the entropy generated by the tube reaches its minimum. For example, in the case of a smooth tube with specified mass flow rate m, heat transfer rate per unit length q', and fluid properties ([rho],[mu],k,T,Pr), this design is represented by the optimum Reynolds number (or optimum tube diameter): Re.sub.opt = 2.023 Pr.sup.0.071.B.sup.0.358/.sub.0 where B.sub.0 is the duty parameter of the tube, B.sub.0 = mq'[rho]]mu].sup.-5/2.(kT).sup.-1/2. This result is valid in the range 10.sup.4 < Re < 10.sup.6 and 0.7 < Pr < 160. As shown by Figure 2, away from this optimum tube size, the entropy generated by the tube increases sharply. The parameter [phi] in Figure 2 is the ratio of the fluid-flow irreversibility divided by the heat-transfer irreversibility.

The competition between heat-transfer and fluid-flow irreversibilities is the reason that the dimensions of many other components can be selected for minimum entropy generation. The optimum sizing of a pin with specified heat transfer rate through the base (q.sub.B), and specified cross flow (U.sub.[infinity].,T.sub.[infinity].,[rho],[omega]) is shown in Figure 3. It permits the calculation of the pinlength L and diameter D, or the respective Reynolds numbers based on U.sub.[infinity]., when the duty parameter B.sub.1 = [rho]v.sup.3.k T.sub.[infinity] / q.sup.2/.sub.B is known. One such chart exists for each value of the dimensionless group M = (k/[lambda]).sup.1/2.Pr.sup.-1/6., where k and [lambda] are the thermal conductivities of the fin and fluid, respectively.

The trade-offs for two subcomponents (Figures 2 and 3) also apply in the design of larger components. The irreversibility of every heat exchanger is also the combined result of the heat-transfer and fluid-flow entropy generation mechanisms.

The opportunity for selecting one or more dimensions of the heat exchanger for minimum overall irreversibility was first pointed out in 1951 by Frank A. McClintock, a professor at M.I.T. Many such trade-offs and optimum sizing procedures have been developed since, especially during the last 10 years. Another example is provided by Figure 4. Plotted on the ordinate is a dimensionless measure N.sub.S of the entropy generated by a compact cross-flow heat exchanger with a ratio of heat capacity flow rates of 0.5 and a ratio of inlet temperatures equal to 0.8. The figure shows that when the number of heat transfer units (N.sub.tu.) is fixed, an optimum geometric aspect ratio 4L/D.sub.h exists for each side of the heat transfer surface. On the abscissa, L represents the length of the flow passage, D.sub.h the hydraulic diameter

More Complex Systems

Finally, an example of an even more complex system that can be designed for minimum entropy generation is the sensible-heat storage element illustrated in Figure 5. The temperature of the storage material varies in cycles, rising during the energy storage phase, dropping during the energy removal phase. During the storage phase, the material is heated by a stream of hot fluid, which is later discharged into the ambient. During the removal phase, the material is cooled by a stream of fluid that is later used in a power plant or in another process. The system is treated as a storer of exergy (or availability), not energy.

Figure 5(b) shows that an optimum duration of the storage phase exists, for which the entropy generation N.sub.S of the complete storage plus removal cycle is minimal. Plotted under the N.sub.S curve is also a breakdown of the overall irreversibility among the various contributing mechanisms. For example, the share labeled N.sub.S,Q.sub.0 represents the entropy generation associated with the dumping of the used hot steam into the ambient during the storage phase. This share plays an increasingly important role when the storage time [theta] is longer than the optimum value for which N.sub.S reaches its minimum. The shares labeled N.sub.S,[delta]T and N.sub.S,[delta]P account for heat-transfer and fluid-flow irreversibilities, respectively. The dimensionless time [theta] is defined as [theta] = mc.sub.P / mc t where mc.sub.p is the capacity rate of the stream, mc is the heat capacity of the entire sotrage material, and t is the time.

The heat exchanger surface between the stream and the storage material of the system of Figure 5 can be optimized (sized) based on the trade-off shown in Figure 4. The new message of Figure 5 is that in the case of an energy storage/removal process, the history of the process can be optimized to reduce its overall irreversibility.

In all the examples in Figures 2-5, the optimization consists of balancing the heat-transfer and fluid-flow irreversibilities. There are, of course, more complicated systems in which the overall entropy generation S.sub.gen is due to more than two competing mechanisms, as in mass exchangers. A nonisothermal mass exchanger has three simultaneous mechanisms of entropy generation: the heat-transfer and fluid-flow irreversibilities already discussed, and the irreversibility due to the transfer of mass in the direction of a finite concentration gradient.

The oldest engineering domain in which designers practiced the minimization of entropy generation and in which their recommendations were built into actual machines is the field of low-temperature refrigeration or cryogenic engineering. In fact, the step-by-step development of efficient refrigeration techniques at progressively lower temperatures can be viewed as a sequence of moves to avoid entropy generation.

The simples example of this is the irreversibility associated with any thermal-insulation feature, say, the leakage of heat along a mechanical support attached to the coldest part of the machine. This irreversibility is of the heat-transfer type. When the geometry of the support is constrained by mechanical considerations, the S.sub.gen value of the support can be reduced substantially by cooling the support along its length, at intermediate temperatures. A simple method of implementing this design consists of using a stream of cold gas that flows along the support toward the warm end.

The minimization of entropy generation is a design philosophy that extends over many sections of power and refrigeration engineering. The minimization of irreversibility is a proper way to apply the principles of thermodynamics and to begin the admittedly complex multidisciplinary tasks demanded by the actual design.
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