Second law analysis: An alternative indicator of system efficiency

Ehergy Vol. 5. pp. iw-873 Pergamm Press Ltd.. MO.

Printed in Great Brilain

SECOND LAW ANALYSIS: AN ALTERNATIVE INDICATOR OF SYSTEM EFFICIENCY HERBERTW. HEVERT R&D Associates, 1401Wilson Blvd, Arlington, VA 22209,U.S.A.

and STEPHENC. HEVERT TRW, Redondo Beach, CA 90278,U.S.A.

Abstract-The most commonly used indicator for the efficiency of energy conversion processes is the ratio of the output of useful energy (heat or work) to the total energy input. This ratio is called fust law efficiency, because it is based on a quantitative accounting of energy which reflects recognition of the first law of thermodynamics and the conservation of energy. Growing awareness of limited energy supplies has prompted renewed interest in conservation. One aspect of this renewed interest has been the search for better indicators of the efficiencies of energy conversion processes. Second law analysis, which examines the efficiency with which available energy is consumed, represents one avenue of this search. As is well known, the second law of thermodynamics defines the availability of energy more restrictively than the first law. Comparisons of tirst and second law efficiencies as measures of effectiveness, described in a wide range of literature on thermodynamics and energy management, are reviewed and summarized in this paper. Principally, first law efficiency is silent on the effectiveness with which availability is consumed. Analyses in terms of the second law of thermodynamics more nearly describe the effectiveness with which systems or processes use available energy. For example, when high quality (low entropy) energy sources are used to provide low quality energy (low grade heat), the waste of potentially useful work is expressed by second law efficiency but not by first law analysis. Many conversion processes used in space heating and cooling are of this nature. Second law analysis allows attribution of losses to design and operating parameters. For example, reducing irreversibility implies larger units of equipment for the same overall heat flow at lower thermal gradients. This introduces tradeoffs between equipment costs and operating costs. Fist law efficiency does not provide enough information to support a comprehensive conservation ethic. Because it measures quantities of work and energy and ignores the quality of energy use, it can tell how much energy is needed to perform a particular task, but not how well that energy is used. It can only support bulk allocation arguments. Second law analysis goes beyond allocation, and provides insights needed to apply energy resources to uses which produce less entropy per unit of useful heat or work. The authors review extant second law analyses of energy conversion components and cycles, and provide some original comparative analyses. These comparisons form the basis for conclusions which augment the traditional first law measure of efficiency by second law analysis. INTRODUCTION

The awareness that widely used energy resources are approaching exhaustion has stimulated interest in applying the second law of thermodynamics to the analysis of energy conversion efficiency. This is not surprising, since traditional first law evaluation is silent on irreversibility. As nonrenewable fuels become more scarce, their ability to produce useful work becomes more precious. Availability, the thermodynamic term for this ability, is characterized in terms of the second law. This renewed interest in second law analysis has produced a range of opinions on its utility and a variety of parameters for its expression. In this paper, we will review some of these arguments and offer our reactions to them as practicing mechanical engineers. While the application of second law analysis is not unique to any particular thermodynamic process, we find the topic of space heating and cooling an appropriate setting for this review for three reasons: (I) It represents a significant area of consumption, since about 60% of the energy used in residential and commercial sectors and about 15% of all energy used in the U.S. is used for space heating and cooling. 865

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(2) It encompasses a wide range of technical options and models for study, especially when combined cycles are included. (3) It provides substantial opportunities for improvement in availability exploitation, because it consists mainly of low-grade energy consumption.

MEASURES

OF EFFECTIVENESS

AND EFFICIENCY

The most commonly used measure for energy conversion efficiency is the ratio of the energy transformed for a useful purpose to the energy input. This ratio is based on the first law of thermodynamics, which states that energy is neither created nor destroyed, but can only be transformed from one form to another. First law efficiency accounts for a fraction of the transformed energy that is useful for a specified purpose. The second law of thermodynamics goes a step further, and reveals the amount of energy that is theoretically useable if we are willing to spend enough time, money, and other resources to make it useful. This theoretically useable energy is called availability. Availability is simply the maximum amount of work done in transferring energy from a source to a receiver within a particular set of surroundings at specified pressure and temperature. Work is regarded as a higher form of energy than heat, because it can be transformed into heat with no loss. This is the practical manifestation of the second law of thermodynamics. Since availability is proportional to the mass or scale of a source, it is said to be an extensive property of the system. Unlike energy, which can only be transformed, availability can be lost in a conversion process. Hence, a waste of availability is a waste of a resource quantity. Unrestrained expansion, heat transfer through a finite temperature difference, and friction are examples of processes that result in the loss of availability. The degradation of availability to a less useful form is known as irreversibility, and represents a loss of ability to do work. Intuitively, something of value is lost when high quality energy is converted into low quality energy with no other product. Even when energy is conserved according to the first law, there is always a loss of energy quality in any real process. For example, when electricity is passed through resistance coils to increase the temperature of a room only a few degrees, first law principles indicate that this process could have an efficiency of 100%. Electricity, however, is a high quality form of energy that has a high potential for conversion to useful work. In this example, it is used to provide low quality heat transfer and wastes much of its availability. Thus, the conversion potential of the electricity has been poorly utilized. A similar reasoning can be applied to the use of chemical energy in gas or oil fired furnaces. Second law analysis also accounts for this loss of quality. Since availability is the maximum amount of useful work that can be obtained from a process, it provides a common denominator between work and heat regardless of quality. Thus, comparisons of non-similar processes for a desired effect can be made with a specific reference to the conversion quality of each process. Second law efficiencies are evaluated on the basis of availability. We have found two basic forms of this parameter. One form accounts for the actual degradation of availability in relation to the input of a process. The other uses the theoretical minimum availability for a given process as the datum. In addition to these second law efficiencies, various other parameters have been proposed. One author uses a grade function to express the availability within an energy source. Another uses a utility function which relates the useful output of a process to the availability consumed. Hamel and Brown’ use three related but essentially different parameters for the second law portion of their analyses of energy systems. The first is a nondimensional thermodynamic variable which characterizes the fraction of energy within a system that is theoretically available. This energy grade function is defined as R=AIE, where E=energy of the system and A=availability. Because it is determined by well defined thermodynamic properties, and because it is insensitive to mass, R gives an instant view of the

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Second law analysis

match between a particular energy source to a particular task in terms of availability. Hamel and Brown’s calculated values for R at TO= 85°F give a sense of proportion to this parameter.

Primary Source

R

electricity natural gas steam (212°F) hot water (150°F) hot air (150°F) air (65%)

1.0 0.913 0.1385 0.00921 0.00596 0.00072

The second parameter from Hamel and Brown is called the utility factor. It applies to the demand element of the system at the point of consumption, and is defined as &~AD

As ’ where A, =availability of the energy supply and A~=availability of the energy discharge. This parameter is analogous to first law efficiency, but it is based on availability instead of energy. For converters, Hamel and Brown use a formulation which is conceptually similar to their utility factor, but which reflects the intermediate status of this element of the system. This they call the availability eficiency

where A”,,,=availability to the user and Ai,=availability of the source. Neither u’ nor E explicitly shows how much availability has been wasted in relation to the theoretical maximum amount of work that could have been produced, although this can readily be caluclated from the same data used to calculate 11’and E. The parameter e, which Reistad* calls effectiveness, is quite similar in concept to Hamel and Brown’s u’ and t E= increase in availability of desired output . decrease in availability required In this case, E is the thermodynamic quality of the output in relation to the thermodynamic quality of the input. Like u’ and E, it does not distinguish wasted availability from the irreversibilities that the second law exacts, even in an ideal system where all losses from mechanical imperfections have been eliminated. These parameters can be powerful design aids. They express how much availability is given up in any conversion or energy end use, irrespective of the theoretical minimum. Selected values of first law efficiency and second law effectiveness from Reistad’s table for typical components (Table 1) show the disparity between parameters for space heating and cooling systems relative to other types of energy conversion. Kreith’ uses a parameter v2 which he calls second Iaw eficiency. Unlike Hamel and Brown and Reistad, Kreith uses the theoretical minimum as his datum

H.W.HEVERT and S.C.

HEVERT

Table 1. Selectedvaluesof first law efficiencv and second law effectiveness for tyoical components. Effectiveness(c)

Efficiency(u)

System

98

Large Electric Generator

96-99

Large Electric Motor

85-95

90

Storage Bpttery

75-90"

80

Large Steam 8oiler

88-92

49

Diesel Engine

30-44

36

Home Gas Furnaceb

60-85

13

Home Oil FurnacB

45-70

11

Steam Electric Generating Plant (Coal Fired)

33-42

36

Home Electric Heat Pumpb

[1ZOP=2.0-4.5]~

Hc+neElectric Resistance Heaterb

*me

17(6.5)'

lOO(38)'

Home Gas Hot Water Haaterb Electric Air Conditionere

60(23)'

17

30-70

17t6.5)'

[COP=2.0-4.0]f

dead state temperature, To, is assumed to be 490"R with '535"wet bulb

for space heating, and 49O"R with 525"R wet bulb for air conditioning. For other processes, the exact value of To is usually not critical, but a value of 510DR ~111 be used. Motes: a)

Usual efficiency reported; really an effectiveness.

b)

The require; heating temperature is assumed to be 590'R.

c)

The value in parenthesis includes the inefficiencyof the electrical generation and transmfssion assuming an efficiency of 38%.

d)

The water is heated to 672OR.

e)

The required cooling temperature Is assumed to be 520"R.

f)

COP is the coefficient of performance.

where Amin=theproduct of the Camot efficiency and quantity of heat transferred [(I-Z’,#)Q] and A =the actual consumption of availability. Kreith’s formula is consistent with the definition of second law efficiency given by the American Institute of Physics,4 which is

where B=available work, or the maximum work performed by a system as it proceeds by any path to a specified final state in thermodynamic equilibrium with the atmosphere. Interaction with the atmosphere is permitted, but work done on the atmosphere is not counted. Ross and Williams’ calculate second law efficiencies according to this formulation, but they give generally lower values than Reistad’s estimates Consumption activity Space Heating Furnace Electric Resistive Air Conditioning Water Heating Gas Electric Electric Power Generation Process Steam Production

Second law efficiency (%) 5.0 2.5 4.5 3.0 1.5 33.0 34.0

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Second law analysis APPLICABILITY OF SECOND LAW ANALYSIS SPACE HEATING AND COOLING

TO

Reistad’s estimates show that, for several processes like the hydraulic turbine and the electric motor, values of 77and e are essentially equal. The reason for this is that when the output of a process is electricity or shaft work, energy is almost entirely available for point-of-consumption use. This high quality energy conversion is reflected in first and second law efficiencies which have nearly equal values. There are many processes, however, where the effectiveness is significantly less than the thermal efficiency. Some of these processes occur in fuel fired boilers, gas furnaces, electric resistance heaters, gas water heaters, electric water heaters, and appliances used for electric cooking and clothes drying. Here, large losses of availability occur in: (1) combustion/heat transfer processes and (2) point-of-consumption utilization. It is also interesting that these processes commonly occur in the residential and commercial sectors of the energy economy. Our society makes extensive use of high grade energy forms for the production of low quality energy. By first law accounting, 1 Btu of energy is wasted for every Btu of energy converted, making for an overall energy efficiency of 50%. For each unit of availability consumed, however, 3 units of energy are lost through irreversibilities.‘j This lower efficiency demonstrates the poor utilization of high quality energy. Electric power plants provide one example of loss of availability at the point of consumption. The availability of a typical fossil fuel is approx. 90% of its internal energy content (0.9E). During power plant combustion, a loss of about 0.25E occurs. Another 0.15E is lost in irreversible heat transfer, making for a total loss of about 0.4E in the combustion process alone. In power generation, another O.lSE is lost. At the point of consumption, however, most equipment makes use of only 0.02-0.05E. The result is an extremely inefficient use of availability, even though electricity is itself a high-quality form of energy. Since the second law is more descriptive than the frrst law with regard to energy conversion and end use, why has it not come into wider practice? We believe there are two essential reasons. First, it does not explicitly provide certain kinds of data which are typically used as a basis for selecting energy equipment. Second, it is difficult to find good matches between the more common energy supplies and certain lower-quality tasks. On the first point, we cannot begin to treat in any depth the complex decision-making process that attends choices for investment in energy equipment. It is widely recognized, however, that these choices are made on a cost-payback basis rather than on an energy conservation basis. First law principles are adequate to support payback analyses. If the economic choice happens to correspond to the least degradation of availability, it is by coincidence and not by design. In general, the most economical choice will not produce the least destruction of availability. As Van Go01 has observed: when heat rates across particular surfaces are increased in order to provide greater force, entropy also increases. To obtain higher forces at specified heat rates per unit of surface area, it is also possible to increase the surface area over which heat is transferred by building bigger equipment and adding to cost. Also, because the manufacture of energy conversion equipment is itself energy intensive, care must be taken to avoid expending more availability in equipment production than would be conserved over its useful life-cycle. The need for larger equipment sizes may be seen by comparing solar heating systems with conventional gas or oil fired heating systems. Solar collectors have much lower heat transfer rates than fuel fired heat sources. Gas or oil tired domestic hot water systems have relatively small storage requirements, perhaps 50 gallons for a family of four. To meet the same heating load, an 80-100 gallon storage capability would be required if a solar heating system were used because of the added warm-up time. The move toward less availability degradation introduces trade-offs between energy efficiency and cost effectiveness. Higher capital investments to reduce the waste of availability will always be limited by the amount that can be justified in fuel cost savings. The other inhibitor to wider use of second law analysis is related to the previous discussion. The paucity of deliverable low-grade energy sources is partly due to the higher capital cost of such equipment. It is also due to the fact that each energy supplier only delivers a quality of energy which corresponds to the highest end use in a demand sector. As a result, most

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H. W. HEVERT and S. C. HEVERT

consumers have no choice but to use a higher quality of energy than necessary because no other is available. Moreover, national and state regulatory processes are typically based on accounting and pricing that correspond to first law principles, and thus provide no check upon this waste of availability. Because of the growing scarcity of fuels which have served us well in performing both high and low grade tasks, it becomes increasingly attractive to consider systems which exploit lower grade energy sources. Those under consideration appear to fall into two main categories: (1) renewable energy converters and (2) waste heat recovery systems. In addition, there is a growing technical interest in replacing end use equipment that has traditionally misused high grade energy with equipment that operates on energy of an appropriate quality. Absorptive chillers, heat pumps, and solar lighting are examples. As fuel shortages loom before us, the old concept of recovering waste heat from higher quality energy conversion to supply lower-grade demands is attracting renewed interest, as is the cogeneration of electricity using excess steam production. The Department of Energy is sponsoring demonstrations on applying these concepts to communities under a program called Integrated Community Energy Systems (ICES).’ We were somewhat surprised to find that thermodynamic analyses of these systems were almost all derived from first law principles, and that their justification came from comparisons with the very poor alternative of all-electric heating and cooling. While modest gains in overall first law efficiency were shown, cost-benefit calculations show no decisive advantage for the integrated systems. Second law analysis, in contrast, could show strikingly better use of availability for ICES, even when compared with the more typical alternatives of using gas or oil for heating and grid power for the electrical load. In the integrated system, 7)2approaches 35%, whereas in the balanced gas-electric system n2 is in the neighborhood of 20%. Turning now from energy supply to energy consumption, we note that air conditioning and refrigeration systems provide significant opportunities for second law effectiveness improvements. The problem with these systems is that they rely upon a quasi-Rankine two phase cycle. Work lost in throttling the regrigerant through expansion valves represents a 20% excess power consumption in most systems. Vapor compression itself is also a highly irreversible process. From the point of view of availability, there is a more attractive option for refrigeration systems in the absorption cycle. This system does not require a compressor, thus eliminating the irreversibilities associated therewith. Heat pumps offer another opportunity to improve second law effectiveness in space heating and cooling. As shown in Table 1, an electric heat pump has an effectiveness of 60% when compared with an electric resistance heater, which has an effectiveness of 17%. The principle demonstrated here is that work is a more valuable form of energy than heat. A resistance heater converts the available energy of electricity from a power plant directly into heat (high to low quality conversion). A heat pump, on the other hand, makes better use of the high availability in electrical energy by using it to transfer heat from one temperature reservoir to another. SECOND LAW ANALYSIS

AS AN INDICATOR

The preceding review of the typical indicators of second law performance in thermodynamic processes and of the application of the second law to space heating and cooling has revealed some of the strengths and limitations of this type of analysis. Our observations are summarized in Table 2. The present basis for energy accounting, and therefore for pricing, regulation, and conservation, is the first law of thermodynamics. Energy management today is founded on the principle that energy resources are best conserved by not using them, resulting in conservation by allocation. At the same time, however, valuable resources are squandered in processes that poorly utilize their true conversion potential. Wastes and losses of energy could also be assessed as wastes and losses of availability. The second law offers a system of energy accounting that takes into account both the quantity consumed and the quality of energy conversion. Our present conservation ethic emphasizes the internal energy of a fuel rather than the system used to convert its chemical energy into useful energy. The result is a greater concern for the consumption or resource quantity than for conversion quality. This basis of energy

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Second law analysis

Table 2. Typical indicators of second law performance in thermodynamic processes. 1st Law Efficiency

2nd Law Efficiency

USEFUL OUTPUT Al-TRIBUTE Accounting for amount of energy.

Specific.

Specific.

Accounting for quality of energy.

Silent.

Specific.

Identificationof thermodynamic limit.

Silent.

Specific.

Applicability to different types of systems.

Misleading when useful output > energy input.

No limit and always (1.

5)

Sensitivity to state of surroundings.

Second order.

First order.

6)

Comparative indicator.

Similar equipment designs.

Less restrictive tasks and surroundings.

7)

Utility to decision makers.

Easy to understand.

Unfamiliar.

8)

Accounting for: cost trade-offs. envimnmental impact, institutional factors, consuner preferences, societal Impacts, preference for other properties of fuels.

Silent.

Silent.

accounting neglects the fact that work is a higher quality form of energy than heat. This stems from system evaluations based primarily on first law analyses, under which work and heat are considered to be equivalent forms of energy. In comparison to first law analysis, the second law of thermodynamics provides the more comprehensive basis for a conservation ethic. It specifically allows conversion processes to be matched to tasks according to energy quality, and this enables identification of conservation strategies based on both the quantity and the quality of energy savings. For example, shipment of 1 Btu of natural gas via a pipeline costs less than transmitting 1 Btu of electricity across the same distance. But this comparison ignores the fact that almost all of the electricity will be available for conversion into work, while only about 25% of the useful energy of the gas will be available to produce the same work. The economics of these options cannot be given a fair comparison without considering both first and second law efficiencies of the supply and the point of consumption. While tist law efficiency is useful in keeping track of energy flows and in comparing devices of a similar type, it has two serious limitations: 0 It assigns high quality forms of energy to low quality purposes. 0 It leads to values greater than unity for air conditioning and refrigeration systems, and it departs from the usual concept of efficiency. Because second law analysis involves the calculation of irreversibilities, ambient conditions are fully taken into account by this measure of efficiency. First law analysis is never so revealing, especially with low grade energy converters and users. Using a textbook example, we calculated the second law efficiency of a compressive refrigeration cycle for the two ambient temperatures-60 and 77”F-and found it to be 52 and 67% respectively. When a system for energy conversion is designed according to first law principles, there is no ultimate datum against which efficiency may be judged. The utility of first law efficiency is limited to comparisons of different systems with the same useful outputs. When a system is designed according to second law principles, we can estimate its highest feasible efficiency under ideal conditions in specified surroundings. In terms of the calculations required, the first law is easier to apply than the second law. It provides direct, useful information for estimating operating costs of alternative systems, since energy suppliers sell units of fuel or electricity and not availability. The first law is insufficient,

H. W. HEVERT and S. C. HEVERI

872

however, to support detailed design tradeoffs among components within thermodynamic processes. Reistad’s example of the coal-fired steam power plant’ shows that energy losses and availability losses do not necessarily occur at the saae points within the system, even when the first and second law system efficiencies are nearly equal. In 1937, Ebaugh said: “The concept of the reversible cycle together with its subsequent developments on available and unavailable energy has enabled the engineer to set up the finest standard of comparison with which to judge real processes. It permits the engineer to design thermal equipment with the full realization of its limitations and its imperfections.” lo Second law analysis reveals far more about the thermodynamic performance of a system than the first law alone. However, it leaves other energy-related questions unanswered. For instance, it does not answer questions pertaining to: cost tradeoffs, environmental impacts of alternative fuels and processes, institutional barriers, consumer preferences, societal impacts, or preferences for other properties of fuels. The particular engineering values of cleaner-burning fuels or of poI tability, for example, have to be dealt with separately, as do institutional constraints, market preferences, and societal impacts.” In summary, second law analysis offers another avenue for identifying new conservation strategies, especially with regard to the exploitation of low-grade energy sources for space heating and cooling and for tradeoff analyses of alternative designs. Second law analysis augments, but does not replace, first law energy balances. We would like to see further evolution of conventions for second law analysis, as we be!ieve this would help to accelerate its wider adoption.

REFERENCES I. Bernard B. Hamel and Harry L. Brown, “Utilization Analysis of Energy Systems”, Energy Sources and Systems Institute, Drexel University, Philadelphia, Pennsylvania. 2. G. M. Reistad. J. Engng Power 439 (1975). 3. Frank Kreith and Jan F. Kreider, Principles of Solar Engineering. McGraw-Hill, New York (1978). 4. “Efficient Use of Energy”, AIP Conf. Prof., No. 25, American Institute of Physics (1975). 5. Marc H. Ross and Robert H. Williams, Bull. Atomic Sci., Nov. (1976). 6. GM. Reistad, J. Engng. Power 432, July (1975). 7. Willem Van Goal, Phys. Today 9, March (1979). 8. “The Community Systems Program, Its Goals and Accomplishments”, Argonne National Laboratory, Energy and Environmental Systems Division, April (1978). 9. G. M. Reistad, J. Engng Power 431, July (1975). 10. Newton C. Ebaugh, Engineering Thermodynamics. Van Nostrand, Amsterdam (1937). I I. M. Greenberg, Proc. 6th Energy Technology Conf., Government Institutes Inc., April (1979).

DISCUSSION In this workshop, our second law analysis is really a combined statement of both the first and second laws. The difference between them is that first law analysis always recognizes a cycle, whereas second law analysis recognizes changes in state. Author’s reponse. Professor Kestin’s comment responds to our introductory remark that much of the literature on second law analysis tends to contradistinguish it from first law analysis. It is often implied that second law analysis is the more satisfactory choice. While second law analysis is a combined statement of the first and the second laws, it produces numerical results which are second law-specific but not first law-specific. If we were allowed only one parameter by which to express the efficiency of a process, we would be compelled to ask which of the two laws gives the more useful information. Fortunately, we are not so strictly bound. In our paper, my co-author and I identified and discussed extant indicators based on both the first and second laws. We observed that indicators of efficiency based on the first law are necessary to support the analysis of specific designs and processes, but not sufficient enough to compare alternative cycles for performing a particular task nor to isolate sources of irreversible losses within particular processes. The second law enables the latter types of analyses. We concluded that both first and second law indicators are useful in engineering practice. After completing our survey, we considered changing (and perhaps should have changed) the title of our paper to “Second Law Analysis-Another important Indicator of System Efficiency”. Joseph Kestin.

Brown

Uniuersity,

U.S.A.

Himonshu V&ii, Generai Efecfric Corporation, U.S.A. In order to provide heating and cooling, we must have low grade energy sources. However, when we concentrate solar energy in making hot water, from a thermodynamic standpoint we are not making it more efficient. Author’s response. The first law efficiency for concentrating collectors is usually higher than that for flat plate collectors. There are two significant reasons for this. First, there is less heat loss area relative to receiver area for concentrating collectors than for flat plate collectors. Secondly, there is a reduction in transient effects, since the thermal mass in a concentrating collector usually is less than that of most Rat plate collectors.

Second law analysis

873

The second law efficiency for a concentrating collector system, however, can indeed be less than the second law efficiency of a flat plate collector system. For most concentrating collectors, the beam component reflected radiation is the primary source of heating, and the effect of diiuse radiation is insignificantt For flat plate collectors, the diffuse component of radiation contributes signiticantly to heating the working fluid. Diffuse radiation, because of scattering, has a higher entropy level than focused beam radiati0n.S Thus the thermodynamic availability, or energy quality, of beam radiation is greater than that of an equivalent quantity of diffuse radiation. With these concepts in mind, we can make a qualitative assessment of comparative thermodynaic efficiency through the following example. Suppose we want to examine the second law efficiency for a water heating system using either a focusing collector or a lIat plate collector. The task for each is to heat a given volume of water from 60 to 140°F.When the water in storage reaches 14CPF,the circulating pump shuts off, and no more heat is absorbed by the storage volume. Each collector must, therefore, deliver a control volume around the flat plate collector, and a control volume around the concentrating collector absorber tube. Since beam radiation has a greater thermodynamic availability than diffuse radiation, the energy input to the concentrating absorber tube is greater than that of the flat plate collector. Using Reistad’s formulation for sedond law effectiveness, we obtain e = increase of availability of output availability input The increase in availability for each system is equal, based on our assumption. The availability input for the focusing system, however, is greater than that of the flat plate system. Thus, r(tIat plate)%(concentrating). In other words, from a task matching point of view, the flat plate collector makes better use of a given input of availability in this application. tJ.A. Dullie and W.A. Beckman, Solar Energy Thermal Processes. Wiley, New York (1974). SF. Kreith and J.F. Kreider, Principles of Solar Engineering. McGraw-Hill, New York (1978).