Thermodynamics 7

A closed system at a given state can attain new states by various means, including work and heat interactions with its surroundings. The exergy value associated with a new state generally differs from the exergy value at the initial state. Using Eq. 7.1, we can determine the change in exergy between the two states. At the initial state E1 1U1 U02 p01V1 V02 T01S1 S02 KE1 PE1 At the final state E2 1U2 U02 p01V2 V02 T01S2 S02 KE2 PE2 Subtracting these we get the exergy change E2 E1 1U2 U12 p01V2 V12 T01S2 S12 1KE2 KE12 1PE2 PE12 (7.3) Note that the dead state values U0, V0, S0 cancel when we subtract the expressions for E1 and E2. Exergy change can be illustrated using Fig. 7.4, which shows an exergy-temperaturepressure surface for a gas together with constant-exergy contours projected on temperature-pressure coordinates. For a system undergoing Process A, exergy increases as its state moves away from the dead state (from 1 to 2). In Process B, exergy decreases as the state moves toward the dead state (from 19 to 29.) exergy change Fig. 7.4 Exergy-temperature-pressure surface for a gas. (a) Three-dimensional view (b) Constant exergy contours on a T-p diagram. Constant-exergy contour Exergy increases Exergy increases +0 2' 1 2 1' B A p (a) (b) T T0 p0 p T E E = 0 at T0, p0 Constantexergy line 7.4 Closed System Exergy Balance Like energy, exergy can be transferred across the boundary of a closed system. The change in exergy of a system during a process would not necessarily equal the net exergy transferred because exergy would be destroyed if irreversibilities were present within the system during the process. The concepts of exergy change, exergy transfer, and exergy destruction are related by the closed system exergy balance introduced in this section. The exergy balance concept is extended to control volumes in Sec. 7.5. Exergy balances are expressions of the second law of thermodynamics and provide the basis for exergy analysis.

Although energy and exergy share common units and exergy transfer accompanies energy transfer, energy and exergy are fundamentally different concepts. Energy and exergy relate, respectively, to the first and second laws of thermodynamics: c Energy is conserved. Exergy is destroyed by irreversibilities. c Exergy expresses energy transfer by work, heat, and mass flow in terms of a common measure—namely, work that is fully available for lifting a weight or, equivalently, as shaft or electrical work.

Although exergy is an extensive property, it is often convenient to work with it on a unit mass or molar basis. Expressing Eq. 7.1 on a unit mass basis, the specific exergy, e, is e 1u u02 p01y y02 T01s s02 V2 /2 gz (7.2) where u, , s, V2 /2, and gz are the specific internal energy, volume, entropy, kinetic energy, and potential energy, respectively, at the state of interest; u0, 0, and s0 are specific properties at the dead state at T0, p0. In Eq. 7.2, the kinetic and potential energies are measured relative to the environment and thus contribute their full values to the exergy magnitude because, in principle, each could be fully converted to work were the system brought to rest at zero elevation relative to the environment. Finally, by inspection of Eq. 7.2, note that the units of specific exergy are the same as for specific energy, kJ/kg or Btu/lb. The specific exergy at a specified state requires properties at that state and at the dead state

Although the cost rates denoted by C# in Eq. 7.30 are evaluated by various means in practice, the present discussion features the use of exergy for this purpose. Since exergy measures the true thermodynamic values of the work, heat, and other interactions between a system and its surroundings as well as the effect of irreversibilities within the system, exergy is a rational basis for assigning costs. With exergy costing, each of the cost rates is evaluated in terms of the associated rate of exergy transfer and a unit cost. Thus, for an entering or exiting stream, we write C# cE # f (7.31) where c denotes the cost per unit of exergy (in $ or cents per kW ? h, for example) and E # f is the associated exergy transfer rate

Another important aspect of thermoeconomics is the use of exergy for allocating costs to the products of a thermal system. This involves assigning to each product the total cost to produce it, namely, the cost of fuel and other inputs plus the cost of owning and operating the system (e.g., capital cost, operating and maintenance costs). Such costing is a common problem in plants where utilities such as electrical power, chilled water, compressed air, and steam are generated in one department and used in others. The plant operator needs to know the cost of generating each utility to ensure that the other departments are charged properly according to the type and amount of each utility used. Common to all such considerations are fundamentals from engineering economics, including procedures for annualizing costs, appropriate means for allocating costs, and reliable cost data

Anticipating the main results of this chapter, exergy is the property that quantifies potential for use. The foregoing example illustrates that, unlike energy, exergy is not conserved but is destroyed by irreversibilities. Subsequent discussion shows that exergy not only can be destroyed by irreversibilities but also can be transferred to and from systems. Exergy transferred from a system to its surroundings without use typically represents a loss. Improved energy resource utilization can be realized by reducing exergy destruction within a system and/or reducing losses. An objective in exergy analysis is to identify sites where exergy destructions and losses occur and rank order them for significance. This allows attention to be centered on aspects of system operation that offer the greatest opportunities for cost-effective improvements.

As in the case of the mass, energy, and entropy balances, the exergy balance can be expressed in various forms that may be more suitable for particular analyses. A convenient form is the closed system exergy rate balance given by dE dt a j a1 T0 Tj bQ # j aW # p0 dV dt b E # d (7.10) where dE/dt is the time rate of change of exergy. The term 11 T0 /Tj2Q # j represents the time rate of exergy transfer accompanying heat transfer at the rate Q # j occurring where the instantaneous temperature on the boundary is Tj . The term W # represents the time rate of energy transfer by work. The accompanying rate of exergy transfer is given by (W # p0dV /dt) where dV/dt is the time rate of change of system volume. The term E # d accounts for the time rate of exergy destruction due to irreversibilities within the system. At steady state, dE/dt 5 dV/dt 5 0 and Eq. 7.10 reduces to give the steady-state form of the exergy rate balance. 0 a j a1 T0 Tj bQ # j W # E # d (7.11a) Note that for a system at steady state, the rate of exergy transfer accompanying the power W # is simply the power. The rate of exergy transfer accompanying heat transfer at the rate Q # j occurring where the temperature is Tj is compactly expressed as E # qj a1 T0 Tj bQ # j (7.12) As shown in the adjacent figure, heat transfer and the accompanying exergy transfer are in the same direction when Tj . T0. steady-state form of the closed system exergy rate balance Alternatively, the exergy destruction can be evaluated using Eq. 7.7 together with the entropy production obtained from an entropy balance. This is left as an exercise. ❶ Recognizing the term (1 2 T0/T) as the Carnot efficiency (Eq. 5.9), the right side of Eq. (b) can be interpreted as the work that could be developed by a reversible power cycle were it to receive energy Q/m at temperature T and discharge energy to the environment by heat transfer at T0. ❷ The right side of Eq. (c) shows that if the system were interacting with the environment, all of the work, W/m, represented by area 1-2-d-a-1 on the p- diagram of Fig. E7.2, would not be fully available for lifting a weight. A portion would be spent in pushing aside the environment at pressure p0. This portion is given by p0(2 2 1), and is represented by area a-b-c-d-a on the p- diagram of Fig. E7.2. ✓Skills Developed Ability to... ❑ evaluate exergy change. ❑ evaluate exergy transfer accompanying heat transfer and work. ❑ evaluate exergy destruction. If heating from saturated liquid to saturated vapor would occur at 1008C (373.15 K), evaluate the exergy transfers accompanying heat transfer and work, each in kJ/kg. Ans. 484, 0. Quick Quiz 1 - T__0 Tj ( ) Eqj = · Q · j Q · j Tj (> T0) 384 Chapter 7 Exergy Analysis Using Eq. 7.12, Eq. 7.11a reads 0 a j E # qj W # E # d (7.11b) In Eqs. 7.11, the rate of exergy destruction within the system, E # d, is related to the rate of entropy production within the system by E # d T0

By applying cost rate balances to the boiler and the turbine, we are able to determine the cost of each product of the cogeneration system. The unit cost of the electricity is determined by Eq. 7.34c and the unit cost of the low-pressure steam is determined by the expression c2 5 c1 together with Eq. 7.32b. The example to follow provides a detailed illustration. The same general approach is applicable for costing the products of a wide-ranging class of thermal systems.

Costing of thermal systems considers costs of owning and operating them. Some observers voice concerns that costs related to the environment often are only weakly taken into consideration in such evaluations. They say companies pay for the right to extract natural resources used in the production of goods and services but rarely pay fully for depleting nonrenewable resources and mitigating accompanying environmental degradation and loss of wildlife habitat, in many cases leaving the cost burden to future generations. Another concern is who pays for the costs of controlling air and water pollution, cleaning up hazardous wastes, and the impacts of pollution and waste on human health—industry, government, the public, or some combination of all three. Yet when agreement about environmental costs is achieved among interested business, governmental, and advocacy groups, such costs are readily integrated in costing of thermal systems, including costing on an exergy basis, which is the present focus

Energy is conserved in every device or process. It cannot be destroyed. Energy entering a system with fuel, electricity, flowing streams of matter, and so on can be accounted for in the products and by-products. However, the energy conservation idea alone is inadequate for depicting some important aspects of resource utilization.

Equation 7.22 shows that the cost of such a loss is less at lower temperatures than at higher temperatures. b b b b b The previous example illustrates what we would expect of a rational costing method. It would not be rational to assign the same economic value for a heat transfer occurring Fig. 7.8 Effect of use temperature Tu on the exergetic efficiency e (Ts 5 2200 K, 5 100%). Heating in industrial furnaces Process steam generation Space heating Tu 0.5 300 540 500 K 900°R 1000 K 1800°R 1500 K 2700°R 1.0 e → 1 (100%) as Tu → Ts e 402 Chapter 7 Exergy Analysis near ambient temperature, where the thermodynamic value is negligible, as for an equal heat transfer occurring at a higher temperature, where the thermodynamic value is significant. Indeed, it would be incorrect to assign the same cost to heat loss independent of the temperature at which the loss is occurring. For further discussion of exergy costing, see Sec. 7.7.3

Exergetic efficiencies are useful for distinguishing means for utilizing fossil fuels that are thermodynamically effective from those that are less so. Exergetic efficiencies also can be used to evaluate the effectiveness of engineering measures taken to improve the performance of systems. This is done by comparing the efficiency values determined before and after modifications have been made to show how much improvement has been achieved. Moreover, exergetic efficiencies can be used to gauge the potential for improvement in the performance of a given system by comparing the efficiency of the system to the efficiency of like systems. A significant difference between these values signals that improved performance is possible. It is important to recognize that the limit of 100% exergetic efficiency should not be regarded as a practical objective. This theoretical limit could be attained only if there were no exergy destructions or losses. To achieve such idealized processes might require extremely long times to execute processes and/or complex devices, both of which are at odds with the objective of cost-effective operation. In practice, decisions are chiefly made on the basis of total costs. An increase in efficiency to reduce fuel consumption, or otherwise utilize fuels better, often requires additional expenditures for facilities and operations. Accordingly, an improvement might not be implemented if an increase in total cost would result. The trade-off between fuel savings and additional investment invariably dictates a lower efficiency than might be achieved theoretically and even a lower efficiency than could be achieved using the best available technology

Exergetic efficiency expressions can take many different forms. Several examples are given in the current section for thermal system components of practical interest. In every instance, the efficiency is derived by the use of the exergy rate balance. The approach used here serves as a model for the development of exergetic efficiency expressions for other components. Each of the cases considered involves a control volume at steady state, and we assume no heat transfer between the control volume and its surroundings. The current presentation is not exhaustive. Many other exergetic efficiency expressions can be written.

Feedwater also enters as saturated liquid, receives exergy by heat transfer from the combustion gases, and exits without temperature change as saturated vapor at a specified condition for use elsewhere. Temperatures of the hot gas and water streams are also shown on the figure

For a compressor or pump operating at steady state with no heat transfer with its surroundings, the exergy rate balance, Eq. 7.17, can be placed in the form aW # cv m # b ef2 ef1 E # d m # Thus, the exergy input to the device, W # cv/m # , is accounted for by the increase in the flow exergy between inlet and exit and the exergy destroyed. The effectiveness of the conversion from work input to flow exergy increase is gauged by the exergetic compressor (or pump) efficienc

For a control volume, the location, types, and true magnitudes of inefficiency and loss can be pinpointed by systematically evaluating and comparing the various terms of 396 Chapter 7 Exergy Analysis the exergy balance for the control volume. This is an extension of exergy accounting, introduced in Sec. 7.4.4. The next two examples provide illustrations of exergy accounting in control volumes. The first involves the steam turbine with stray heat transfer considered previously in Example 6.6, which should be quickly reviewed before studying the current example

For a turbine operating at steady state with no heat transfer with its surroundings, the steady-state form of the exergy rate balance, Eq. 7.17, reduces as follows: 0 a j a1 T0 Tj b 0 Q # j W # cv m # 1ef1 ef22 E # d This equation can be rearranged to read ef1 ef2 W # cv m # E # d m # (7.23) Traditional oil reserves are widely anticipated to decline in years ahead. But the impact could be lessened if cost-effective and environmentally benign technologies can be developed to recover oil-like substances from abundant oil shale and oil sand deposits in the United States and Canada. Production means available today are both costly and inefficient in terms of exergy demands for the blasting, digging, transporting, crushing, and heating of the materials rendered into oil. Current production means not only use natural gas and large amounts of water but also may cause wide-scale environmental devastation, including air and water pollution and significant amounts of toxic waste. Moreover, pipeline delivery of Canadian oil sands to the United States has been controversial. Some say that oil sands corrode or damage pipelines more than conventional crude oil, making pipeline failure more likely. Others disagree that oil sands necessarily present significant environmental danger. Although significant rewards await developers of improved technologies, the challenges also are significant. Some say efforts are better directed to using traditional oil reserves more efficiently and to developing alternatives to oil-based fuels such as cellulosic ethanol produced with relatively lowcost biomass from urban, agricultural, and forestry sources. Oil from Shale and Sand Deposits—The Jury Is Still Out HORIZONS 7.6 Exergetic (Second Law) Efficiency 403 The term on the left of Eq. 7.23 is the decrease in flow exergy from turbine inlet to exit. The equation shows that the flow exergy decrease is accounted for by the turbine work developed, W # cv /m # , and the exergy destroyed, E # d /m # . A parameter that gauges how effectively the flow exergy decrease is converted to the desired product is the exergetic turbine efficiency e W # cv /m # ef1 ef 2 (7.24) This particular exergetic efficiency is sometimes referred to as the turbine effectiveness. Carefully note that the exergetic turbine efficiency is defined differently from the isentropic turbine efficiency introduced in Sec. 6.1

For simplicity, we assume the feedwater and combustion air enter the boiler with negligible exergy and cost. Thus, Eq. 7.30 reduces as follows: C # 1 C # F C # 0 a C # 0 w Z # b Then, with Eq. 7.31 we get c1E # f1 cFE # f F Z # b (7.32a) Solving for c1, the unit cost of the high-pressure steam is c1 cF a E # f F E # f1 b Z # b E # f1 (7.32b) This equation shows that the unit cost of the high-pressure steam is determined by two contributions related, respectively, to the cost of the fuel and the cost of owning and operating the boiler. Due to exergy destruction and loss, less exergy exits the boiler with the high-pressure steam than enters with the fuel. Thus, E # fF/E # f1 is invariably greater than one, and the unit cost of the high-pressure steam is invariably greater than the unit cost of the fuel.

For the heat pump of Examples 6.8 and 6.14, determine the exergy destruction rates, each in kW, for the compressor, condenser, and throttling valve. If exergy is valued at $0.08 per kW h, determine the daily cost of electricity to operate the compressor and the daily cost of exergy destruction in each component. Let T0 5 273 K (08C), which corresponds to the temperature of the outside air.

For thermodynamic analysis involving the exergy concept, it is necessary to model the atmosphere used in the foregoing discussion. The resulting model is called the exergy reference environment, or simply the environment. In this book the environment is regarded to be a simple compressible system that is large in extent and uniform in temperature, T0, and pressure, p0. In keeping with the idea that the environment represents a portion of the physical world, the values for both p0 and T0 used throughout a particular analysis are normally taken as typical ambient conditions, such as 1 atm and 258C (778F). Additionally, the intensive properties of the environment do not change significantly as a result of any process under consideration, and the environment is free of irreversibilities. When a system of interest is at T0 and p0 and at rest relative to the environment, we say the system is at the dead state. At the dead state there can be no interaction between system and environment and, thus, no potential for developing work.

From our study of heat transfer, we know an inverse relation exits between DTave and the boiler tube surface area required for a desired heat transfer rate between the streams. For example, if we design for a small average temperature difference to reduce exergy destruction within the heat exchanger, this dictates a large surface area and typically a more costly boiler. From such considerations, we infer that boiler capital cost increases with decreasing DTave. This variation is shown in Fig. 7.13, again on an annualized basis.

In this section, the exergy balance is extended to a form applicable to control volumes at steady state. The control volume form is generally the most useful for engineering analysis. The exergy rate balance for a control volume can be derived using an approach like that employed in the box of Sec. 4.1, where the control volume form of the mass rate balance is obtained by transforming the closed system form. However, as in the developments of the energy and entropy rate balances for control volumes (Secs. 4.4.1 and 6.9, respectively), the present derivation is conducted less formally by modifying 388 Chapter 7 Exergy Analysis the closed system rate form, Eq. 7.10, to account for the exergy transfers at the inlets and exits. The result is

In this section, we list five important aspects of the exergy concept: 1. Exergy is a measure of the departure of the state of a system from that of the environment. It is therefore an attribute of the system and environment together. However, once the environment is specified, a value can be assigned to exergy in terms of property values for the system only, so exergy can be regarded as a property of the system. Exergy is an extensive property. 2. The value of exergy cannot be negative. If a system were at any state other than the dead state, the system would be able to change its condition spontaneously toward the dead state; this tendency would cease when the dead state was reached. No work must be done to effect such a spontaneous change. Accordingly, any change in state of the system to the dead state can be accomplished with at least zero work being developed, and thus the maximum work (exergy) cannot be negative. 3. Exergy is not conserved but is destroyed by irreversibilities. A limiting case is when exergy is completely destroyed, as would occur if a system were permitted to undergo a spontaneous change to the dead state with no provision to obtain work. The potential to develop work that existed originally would be completely wasted in such a spontaneous process. 4. Exergy has been viewed thus far as the maximum theoretical work obtainable from an overall system of system plus environment as the system passes from a given state to the dead state. Alternatively, exergy can be regarded as the magnitude of the minimum theoretical work input required to bring the system from the dead state to the given state. Using energy and entropy balances as above, we can readily develop Eq. 7.1 from this viewpoint. This is left as an exercise. 5. When a system is at the dead state, it is in thermal and mechanical equilibrium with the environment, and the value of exergy is zero. More precisely, the thermomechanical contribution to exergy is zero. This modifying term distinguishes the exergy concept of the present chapter from another contribution to exergy introduced in Sec. 13.6, where the contents of a system at the dead state are permitted to enter into chemical reaction with environmental components and in so doing develop additional work. This contribution to exergy is called chemical exergy. The chemical exergy concept is important in the second law analysis of many types of systems, in particular systems involving combustion. Still, as shown in this chapter, the thermomechanical exergy concept suffices for a wide range of thermodynamic evaluations.

Instead of the body cooling spontaneously as considered in Fig. 5.1a, Fig. 7.2 shows that if the heat transfer Q during cooling is passed to the power cycle, work Wc can be developed, while Q0 is discharged to the atmosphere. These are the only energy transfers. The work Wc is fully available for lifting a weight or, equivalently, as shaft work or electrical work. Ultimately the body cools to T0, and no more work would be developed. At equilibrium, the body and atmosphere each possess energy, but there no longer is any potential for developing work from the two because no further interaction can occur between them.

Is costing an art or a science? The answer is a little of both. Cost engineering is an important engineering subdiscipline aimed at objectively applying real-world thermoeconomics 406 Chapter 7 Exergy Analysis costing experience in engineering design and project management. Costing services are provided by practitioners skilled in the use of specialized methodologies, cost models, and databases, together with costing expertise and judgment garnered from years of professional practice. Depending on need, cost engineers provide services ranging from rough and rapid estimates to in-depth analyses. Ideally, cost engineers are involved with projects from the formative stages, for the output of cost engineering is an essential input to decision making. Such input can be instrumental in identifying feasible options from a set of alternatives and even pinpointing the best option

Let us begin by evaluating the cost of the high-pressure steam produced by the boiler. For this, we consider a control volume enclosing the boiler. Fuel and air enter the boiler separately and combustion products exit. Feedwater enters and high-pressure steam exits. The total cost to produce the exiting high-pressure steam equals the total cost of the entering streams plus the cost of owning and operating the boiler. This is expressed by the following cost rate balance for the boiler C# 1 C # F C # a C # w Z # b (7.30) where C# is the cost rate of the respective stream (in $ per hour, for instance). Z# b accounts for the cost rate associated with owning and operating the boiler, including expenses related to proper disposal of the combustion products. In the present discussion, the cost rate Z# b is presumed known from a previous economic analysis

Most thermal systems are supplied with exergy inputs derived directly or indirectly from the consumption of fossil fuels. Accordingly, avoidable destructions and losses of exergy represent the waste of these resources. By devising ways to reduce such inefficiencies, better use can be made of fuels. The exergy balance can be applied to determine the locations, types, and true magnitudes of energy resource waste and thus can play an important part in developing strategies for more effective fuel use. In Example 7.3, the steady-state form of the closed system energy and exergy rate balances are applied to an oven wall to evaluate exergy destruction and exergy loss, which are interpreted in terms of fossil fuel use.

Next, consider a control volume enclosing the turbine. The total cost to produce the electricity and low-pressure steam equals the cost of the entering high-pressure steam plus the cost of owning and operating the device. This is expressed by the cost rate balance for the turbine C# e C # 2 C # 1 Z # t (7.33) where C# e is the cost rate associated with the electricity, C# 1 and C# 2 are the cost rates associated with the entering and exiting steam, respectively, and Z# t accounts for the cost rate associated with owning and operating the turbine. With exergy costing, each cost rate balance exergy unit cost 410 Chapter 7 Exergy Analysis of the cost rates C# e , C# 1 , and C# 2 is evaluated in terms of the associated rate of exergy transfer and a unit cost. Equation 7.33 then appears as ceW # e c2E # f2 c1E # f1 Z # t (7.34a) The unit cost c1 in Eq. 7.34a is given by Eq. 7.32b. In the present discussion, the same unit cost is assigned to the low-pressure steam; that is, c2 5 c1. This is done on the basis that the purpose of the turbine is to generate electricity, and thus all costs associated with owning and operating the turbine should be charged to the power generated. We can regard this decision as a part of the cost accounting considerations that accompany the thermoeconomic analysis of thermal systems. With c2 5 c1, Eq. 7.34a becomes ceW # e c11E # f1 E # f22 Z # t (7.34b) The first term on the right side accounts for the cost of the exergy used and the second term accounts for the cost of owning and operating the system. Solving Eq. 7.34b for ce, and introducing the exergetic turbine efficiency e from Eq. 7.24 ce c1 e Z # t W # e (7.34c) This equation shows that the unit cost of the electricity is determined by the cost of the high-pressure steam and the cost of owning and operating the turbine. Because of exergy destruction within the turbine, the exergetic efficiency is invariably less than one; therefore, the unit cost of electricity is invariably greater than the unit cost of the high-pressure steam.

Note that work Wc also could be developed by the system of Fig. 7.2 if the initial temperature of the body were less than that of the atmosphere: Ti , T0. In such a case, the directions of the heat transfers Q and Q0 shown on Fig. 7.2 would each reverse. Work could be developed as the body warms to equilibrium with the atmosphere. Since there is no net change of state for the power cycle of Fig. 7.2, we conclude that the work Wc is realized solely because the initial state of the body differs from that of the atmosphere. Exergy is the maximum theoretical value of such work.

Returning to Fig. 7.1, note that the fuel present initially has economic value while the final slightly warm mixture has little value. Accordingly, economic value decreases in this process. From such considerations we might infer there is a link between exergy and economic value, and this is the case as we will see in subsequent discussions.

Tasks such as space heating, heating in industrial furnaces, and process steam generation commonly involve the combustion of coal, oil, or natural gas. When the products of combustion are at a temperature significantly greater than required by a given task, the end use is not well matched to the source and the result is inefficient use of the fuel burned. To illustrate this simply, refer to Fig. 7.7, which shows a closed system receiving a heat transfer at the rate Q # s at a source temperature Ts and delivering Q # u at a use temperature Tu. Energy is lost to the surroundings by heat transfer at a rate Q # l across a portion of the surface at Tl. All energy transfers shown on the figure are in the directions indicated by the arrows. Assuming that the system of Fig. 7.7 operates at steady state and there is no work, the closed system energy and exergy rate balances Eqs. 2.37 and 7.10 reduce, respectively, to dE dt 0 1Q # s Q # u Q # l2 W # 0 dE dt 0 c a1 T0 Ts bQ # s a1 T0 Tu bQ # u a1 T0 Tl bQ # l d cW # 0 p0 dV dt 0 d E # d These equations can be rewritten as follows Q # s Q # u Q # 1 (7.19a) a1 T0 Ts bQ # s a1 T0 Tu bQ # u a1 T0 Tl bQ # l E # d (7.19b) Equation 7.19a indicates that the energy carried in by heat transfer, Q # s, is either used, Q # u , or lost to the surroundings, Q # l . This can be described by an efficiency in terms of energy rates in the form product/input as h Q # u Q # s (7.20) In principle, the value of can be increased by applying insulation to reduce the loss. The limiting value, when Q # l 0, is h 5 1 (100%). Equation 7.19b shows that the exergy carried into the system accompanying the heat transfer Q # s is either transferred from the system accompanying the heat transfers Q # u and Q # l or destroyed by irreversibilities within the system. This can be described by an efficiency in terms of exergy rates in the form product/input as e 11 T0 /Tu2Q # u 11 T0 /Ts2Q # s (7.21a) Introducing Eq. 7.20 into Eq. 7.21a results in e h a 1 T0 /Tu 1 T0 /Ts b (7.21b) The parameter e, defined with reference to the exergy concept, may be called an exergetic efficiency. Note that h and e each gauge how effectively the input is converted to the product. The parameter h does this on an energy basis, whereas e does it on an exergy basis. As discussed next, the value of e is generally less than unity even when h 5 1. Equation 7.21b indicates that a value for h as close to unity as practical is important for proper utilization of the exergy transferred from the hot combustion gas to the system. However, this alone would not ensure effective utilization. The temperatures

The actual design process differs significantly from the simple case considered here. For one thing, costs cannot be determined as precisely as implied by the curves in Fig. 7.13. Fuel prices vary widely over time, and equipment costs may be difficult to predict as they often depend on a bidding procedure. Equipment is manufactured in discrete sizes, so the cost also would not vary continuously as shown in the figure. Furthermore, thermal systems usually consist of several components that interact with one another. Optimization of components individually, as considered for the heat exchanger unit of the boiler, does not guarantee an optimum for the overall system. Finally, the example involves only DTave as a design variable. Often, several design variables must be considered and optimized simultaneously

The closed system exergy balance is given by Eq. 7.4a. See the box for its development. E2 E1 2 1 a1 T0 Tb bdQ W p01V2 V12 T0s (7.4a) For specified end states and given values of p0 and T0, the exergy change E2 2 E1 on the left side of Eq. 7.4a can be evaluated from Eq. 7.3. The underlined terms on the right depend explicitly on the nature of the process, however, and cannot be determined by knowing only the end states and the values of p0 and T0. These terms are interpreted in the discussions of Eqs. 7.5-7.7, respectively

The direct contact heat exchanger shown in Fig. 7.11 operates at steady state with no heat transfer with its surroundings. The exergy rate balance, Eq. 7.13a, reduces to 0 a j a1 T0 Tj b 0 Q # j W # 0 cv m # 1ef1 m # 2ef22m # 3ef3 E # d With m # 3 m # 1 m # 2 from a mass rate balance, this can be written as m # 11ef1 ef32 m # 21ef3 ef22 E # d (7.28) The term on the left of Eq. 7.28 accounts for the decrease in the exergy of the hot stream between inlet and exit. The first term on the right accounts for the increase in the exergy of the cold stream between inlet and exit. Regarding the hot stream as supplying the exergy increase of the cold stream as well as the exergy destroyed by irreversibilities, we can write an exergetic efficiency for a direct contact heat exchanger as

The discussion to this point of the current section can be summarized by the following definition of exergy: Exergy is the maximum theoretical work obtainable from an overall system consisting of a system and the environment as the system comes into equilibrium with the environment (passes to the dead state). Interactions between the system and the environment may involve auxiliary devices, such as the power cycle of Fig. 7.2, that at least in principle allow the realization of the work. The work developed is fully available for lifting a weight or, equivalently, as shaft work or electrical work. We might expect that the maximum theoretical work would be obtained when there are no irreversibilities. As considered in the next section, this is the case.

The first underlined term on the right side of Eq. 7.4a is associated with heat transfer to or from the system during the process. It is interpreted as the exergy transfer accompanying heat transfer. That is, Eq £ exergy transfer accompanying heat transfer § 2 1 a1 T0 Tb bdQ (7.5) where Tb denotes the temperature on the boundary where heat transfer occurs. The second underlined term on the right side of Eq. 7.4a is associated with work. It is interpreted as the exergy transfer accompanying work. That is, Ew c exergy transfer accompanying work d 3W p01V2 V124 (7.6) The third underlined term on the right side of Eq. 7.4a accounts for the destruction of exergy due to irreversibilities within the system. It is symbolized by Ed. That is Ed T0s (7.7) With Eqs. 7.5, 7.6, and 7.7, Eq. 7.4a is expressed alternatively as E2 E1 Eq Ew Ed (7.4b) Although not required for the practical application of the exergy balance in any of its forms, exergy transfer terms can be conceptualized in terms of work, as for the exergy concept itself. See the box for discussion.

The following examples illustrate the use of mass, energy, and exergy rate balances for the evaluation of exergy destruction in control volumes at steady state. Property Fig. 7.6 Comparing energy and exergy for a control volume at steady state. (a) Energy analysis. (b) Exergy analysis. 90 MW (heat transfer) 10 MW (at inlet i) 100 MW Energy In Q · cv = 90 MW W · = 40 MW 40 MW (power) 60 MW (at exit e) 100 MW Energy Out (a) (b) 60 MW (heat transfer) 2 MW (at inlet i) 62 MW Exergy In Exergy Destroyed = 62 MW - 55 MW = 7 MW 40 MW (power) 15 MW (at exit e) 55 MW Exergy Out V2 --2 m · i [h + + gz]i = 10 MW V2 --2 m · e [h + + gz]e = 60 MW i e E · q = 60 MW E · d = 7 MW W · = 40 MW E · fi = 2 MW E · fe = 15 MW i e 7.5 Exergy Rate Balance for Control Volumes at Steady State 391 data also play an important role in arriving at solutions. The first example involves the expansion of steam through a valve (a throttling process, Sec. 4.10). From an energy perspective, the expansion occurs without loss. Yet, as shown in Example 7.5, such a valve is a site of inefficiency quantified thermodynamically in terms of exergy destruction.

The heat exchanger shown in Fig. 7.10 operates at steady state with no heat transfer with its surroundings and both streams at temperatures above T0. The exergy rate balance, Eq. 7.13a, reduces to 0 a j a1 T0 Tj b 0 Q # j W # 0 cv 1m # hef1 m # cef3221m # hef2 m # cef42 E # d where m # h is the mass flow rate of the hot stream and m # c is the mass flow rate of the cold stream. This can be rearranged to read m # h1ef1 ef22 m # c1ef4 ef32 E # d (7.26) The term on the left of Eq. 7.26 accounts for the decrease in the exergy of the hot stream. The first term on the right accounts for the increase in exergy of the cold stream. Regarding the hot stream as supplying the exergy increase of the cold stream as well as the exergy destroyed, we can write an exergetic heat exchanger efficiency as

The introduction to the second law in Chap. 5 provides a basis for the exergy concept, as considered next. Principal conclusions of the discussion of Fig. 5.1 given on p. 242 are that c a potential for developing work exists whenever two systems at different states are brought into communication, and c work can be developed as the two systems are allowed to come into equilibrium. In Fig. 5.1a, for example, a body initially at an elevated temperature Ti placed in contact with the atmosphere at temperature T0 cools spontaneously. To conceptualize how work might be developed in this case, see Fig. 7.2. The figure shows an overall system with three elements: the body, a power cycle, and the atmosphere at T0 and p0. The atmosphere is presumed to be large enough that its temperature and pressure remain constant. Wc denotes the work of the overall system

The objective of this section is to show the use of the exergy concept in assessing the effectiveness of energy resource utilization. As part of the presentation, the exergetic efficiency concept is introduced and illustrated. Such efficiencies are also known as second law efficiencies.

The present discussion centers on the heat exchanger unit. Let us think about its total cost as the sum of fuel-related and capital costs. We will also take the average temperature difference between the two streams, DTave, as the design variable. From our study of the second law of thermodynamics, we know that the average temperature difference between the two streams is a measure of exergy destruction associated with heat transfer between them. The exergy destroyed owing to heat transfer originates in the fuel entering the boiler. Accordingly, a cost related to fuel consumption can be attributed to this source of irreversibility. Since exergy destruction increases with temperature difference between the streams, the fuel-related cost increases with increasing DTave. This variation is shown in Fig. 7.13 on an annualized basis, in dollars per year

The steady-state exergy rate balance, Eq. 7.13a, can be expressed more compactly as Eq. 7.13b 0 a j E # q j W # cv a i E # fi ae E # fe E # d (7.13b) where E # q j a1 T0 Tj b Q # j (7.15) E # fi m # iefi (7.16a) E # fe m # eefe (7.16b) are exergy transfer rates. Equation 7.15 has the same interpretation as given for Eq 7.5 in the box on p. 380, only on a time rate basis. Also note that at steady state the rate of exergy transfer accompanying the power W # cv is simply the power. Finally, the rate of exergy destruction within the control volume E # d is related to the rate of entropy production by T0s # cv . If there is a single inlet and a single exit, denoted by 1 and 2, respectively, the steady-state exergy rate balance, Eq. 7.13a, reduces to 0 a j a1 T0 Tj b Q # j W # cv m # 1ef1 ef22 E # d (7.17) where m # is the mass flow rate. The term (ef1 2 ef2) is evaluated using Eq. 7.14 as

There are two main sources of exergy destruction in the boiler: (1) irreversible heat transfer occurring between the hot combustion gases and the water flowing through the boiler tubes, and (2) the combustion process itself. To simplify the present discussion, the boiler is considered to consist of a combustor unit in which fuel and air are burned to produce hot combustion gases, followed by a heat exchanger unit where water is vaporized as the hot gases cool

Thermal systems typically experience significant work and/or heat interactions with their surroundings, and they can exchange mass with their surroundings in the form of hot and cold streams, including chemically reactive mixtures. Thermal systems appear in almost every industry, and numerous examples are found in our everyday lives. Their design and operation involve the application of principles from thermodynamics, fluid mechanics, and heat transfer, as well as such fields as materials, manufacturing, and mechanical design. The design and operation of thermal systems also require explicit consideration of engineering economics, for cost is always a consideration. The term thermoeconomics may be applied to this general area of application, although it is often applied more narrowly to methodologies combining exergy and economics for optimization studies during design of new systems and process improvement of existing systems

To explore further the costing of thermal systems, consider the simple cogeneration system operating at steady state shown in Fig. 7.14. The system consists of a boiler and a turbine, with each having no significant heat transfer to its surroundings. The figure is labeled with exergy transfer rates associated with the flowing streams, where the subscripts F, a, P, and w denote fuel, combustion air, combustion products, and feedwater, respectively. The subscripts 1 and 2 denote high- and lowpressure steam, respectively. Means for evaluating the exergies of the fuel and combustion products are introduced in Chap. 13. The cogeneration system has two principal products: electricity, denoted by W # e, and low-pressure steam for use in some process. The objective is to determine the cost at which each product is generated.

To illustrate the use of exergy reasoning in design, consider Fig. 7.12 showing a boiler at steady state. Fuel and air enter the boiler and react to form hot combustion gases.

To summarize, in each of its forms Eq. 7.4 states that the change in exergy of a closed system can be accounted for in terms of exergy transfers and the destruction of exergy due to irreversibilities within the system. When applying the exergy balance, it is essential to observe the requirements imposed by the second law on the exergy destruction: In accordance with the second law, the exergy destruction is positive when irreversibilities are present within the system during the process and vanishes in the limiting case where there are no irreversibilities. That is, Ed: e 0 0 irreversibilities present within the system no irreversibilities present within the system (7.8) The value of the exergy destruction cannot be negative. Moreover, exergy destruction is not a property. On the other hand, exergy is a property, and like other properties, the change in exergy of a system can be positive, negative, or zero: E2 E1 : • 0 0 0 For an isolated system, no heat or work interactions with the surroundings occur, and thus there are no transfers of exergy between the system and its surroundings. Accordingly, the exergy balance reduces to give ¢E4isol Ed4isol (7.9) Since the exergy destruction must be positive in any actual process, the only processes of an isolated system that occur are those for which the exergy of the isolated system decreases. For exergy, this conclusion is the counterpart of the increase of entropy principle (Sec. 6.8.1) and, like the increase of entropy principle, can be regarded as an alternative statement of the second law. In Example 7.2, we consider exergy change, exergy transfer, and exergy destruction for the process of water considered in Example 6.1, which should be quickly reviewed before studying the current example.

Ts and Tu are also important, with exergy utilization improving as the use temperature Tu approaches the source temperature Ts. For proper utilization of exergy, therefore, it is desirable to have a value for as close to unity as practical and also a good match between the source and use temperatures. To emphasize further the central role of the use temperature, a graph of Eq. 7.21b is provided in Fig. 7.8. The figure gives the exergetic efficiency e versus the use temperature Tu for an assumed source temperature Ts 5 2200 K (39608R). Figure 7.8 shows that e tends to unity (100%) as the use temperature approaches Ts. In most cases, however, the use temperature is substantially below Ts. Indicated on the graph are efficiencies for three applications: space heating at Tu 5 320 K (5768R), process steam generation at Tu 5 480 K (8648R), and heating in industrial furnaces at Tu 5 700 K (12608R). These efficiency values suggest that fuel is used far more effectively in highertemperature industrial applications than in lower-temperature space heating. The especially low exergetic efficiency for space heating reflects the fact that fuel is consumed to produce only slightly warm air, which from an exergy perspective has little utility. The efficiencies given on Fig. 7.8 are actually on the high side, for in constructing the figure we have assumed h to be unity (100%). Moreover, as additional destruction and loss of exergy are associated with combustion, the overall efficiency from fuel input to end use would be much less than indicated by the values shown on the figure. Costing Heat Loss For the system in Fig. 7.7, it is instructive to consider further the rate of exergy loss accompanying the heat loss Q # l ; that is, 11 T0 /Tl2Q # l . This expression measures the true thermodynamic value of the heat loss and is graphed in Fig. 7.9. The figure shows that the value of the heat loss in terms of exergy depends significantly on the temperature at which the heat loss occurs. We might expect that the economic value of such a loss varies similarly with temperature, and this is the case.

The total cost is the sum of the capital cost and the fuel cost. The total cost curve shown in Fig. 7.13 exhibits a minimum at the point labeled a. Notice, however, that the curve is relatively flat in the neighborhood of the minimum, so there is a range of DTave values that could be considered nearly optimal from the standpoint of minimum total cost. If reducing the fuel cost were deemed more important than minimizing the

cThe total cost is the sum of the capital cost and the fuel cost. The total cost curve shown in Fig. 7.13 exhibits a minimum at the point labeled a. Notice, however, that the curve is relatively flat in the neighborhood of the minimum, so there is a range of DTave values that could be considered nearly optimal from the standpoint of minimum total cost. If reducing the fuel cost were deemed more important than minimizing the

capital cost, we might choose a design that would operate at point a9. Point a0 would be a more desirable operating point if capital cost were of greater concern. Such trade-offs are common in design situations.

he exergy of a system, E, at a specified state is given by the expression E 1U U02 p01V V02 T01S S02 KE PE (7.1) where U, KE, PE, V, and S denote, respectively, internal energy, kinetic energy, potential energy, volume, and entropy of the system at the specified state. U0, V0, and S0 denote internal energy, volume, and entropy, respectively, of the system when at the dead state. In this chapter kinetic energy and potential energy are each evaluated relative to the environment. Thus, when the system is at the dead state, it is at rest relative to the environment and the values of its kinetic and potential energies are zero: KE0 5 PE0 5 0. By inspection of Eq. 7.1, the units of exergy are seen to be the same as those of energy. Equation 7.1 can be derived by applying energy and entropy balances to the overall system shown in Fig. 7.3 consisting of a closed system and an environment. See the box for the derivation of Eq. 7.1.

where the underlined terms account for exergy transfer where mass enters and exits the control volume, respectively. At steady state, dEcv/dt 5 dVcv/dt 5 0, giving the steady-state exergy rate balance 0 a j a1 T0 Tj bQ # j W # cv a i m # iefi ae m # eefe E # d (7.13a) where efi accounts for the exergy per unit of mass entering at inlet i and efe accounts for the exergy per unit of mass exiting at exit e. These terms, known as the specific flow exergy, are expressed as ef h h0 T01s s02 V2 2 gz (7.14) where h and s represent the specific enthalpy and entropy, respectively, at the inlet or exit under consideration; h0 and s0 represent the respective values of these properties when evaluated at T0, p0. See the box for a derivation of Eq. 7.14 and discussion of the flow exergy concept.

Thermodynamics 7

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