Thermodynamics 13

Analyzing Closed Systems Let us consider next a closed system involving a combustion process. In the absence of kinetic and potential energy effects, the appropriate form of the energy balance is UP UR Q W where UR denotes the internal energy of the reactants and UP denotes the internal energy of the products. If the reactants and products form ideal gas mixtures, the energy balance can be expressed as a P nu a R nu Q W (13.16) where the coefficients n on the left side are the coefficients of the reaction equation giving the moles of each reactant or product. Since each component of the reactants and products behaves as an ideal gas, the respective specific internal energies of Eq. 13.16 can be evaluated as u h RT, so the equation becomes Q W a P n1h RTP2 a R n1h RTR2 (13.17a) where TP and TR denote the temperature of the products and reactants, respectively. With expressions of the form of Eq. 13.13 for each of the reactants and products, Eq. 13.17a can be written alternatively as Q W a P n1hf ¢h RTP2 a R n1hf ¢h RTR2 a P n1hf ¢h2 a R n1hf ¢h2 RTP a P n RTR a R n (13.17b) The enthalpy of formation terms are obtained from Table A-25 or Table A-25E. The ¢h terms are evaluated from Table A-23 or Table A-23E. The foregoing concepts are illustrated in Example 13.6, where a gaseous mixture burns in a closed, rigid container

As suggested by this discussion, IT is also useful for analyzing reacting systems. In particular, the equation solver and property retrieval features of IT allow the adiabatic flame temperature to be determined without the iteration required when using table data. In Example 13.8, we show how the adiabatic flame temperature can be determined iteratively using table data or Interactive Thermodynamics: IT.

By paralleling the development given in Sec. 13.7.1 leading to Eq. 13.44b, we can in principle determine the standard chemical exergy of any substance not present in the environment. With such a substance playing the role of CaHb in the previous development, we consider a reaction of the substance with other substances for which the standard chemical exergies are known, and write ech ¢G a P nech a R nech (13.45) where G is the change in Gibbs function for the reaction, regarding each substance as separate at temperature T0 and pressure p0. The underlined term corresponds to the underlined term of Eq. 13.44b and is evaluated using the known standard chemical exergies, together with the n's giving the moles of these reactants and products per mole of the substance whose chemical exergy is being evaluated. consider the case of ammonia, NH3, and T0 5 298.15 K (25C), p0 5 1 atm. Letting NH3 play the role of CaHb in the development leading to Eq. 13.44b, we can consider any reaction of NH3 with other substances for which the standard chemical exergies are known. For the reaction NH3 3 4O2 S 1 2N2 3 2H2O1l2 Eq. 13.45 takes the form ech NH3 3gNH3 3 4 gO2 1 2 gN2 3 2 gH2O1l241T0, p02 1 2 ech N2 3 2 ech H2O1l2 3 4 ech O2 As expected, this agrees closely with the value listed for octane in Table A-26 (Model II): 5,413,100 kJ/kmol. Dividing by the molecular weight, the chemical exergy is obtained on a unit mass basis: ech 5,413,705 114.22 47,397 kJ/kg The chemical exergies determined with the two approaches used in parts (a) and (b) also closely agree. ❶ A molar analysis of this environment on a dry basis reads: O2: 21%, N2, CO2 and the other dry components: 79%. This is consistent with the dry air analysis used throughout the chapter. The water vapor present in the assumed environment corresponds to the amount of vapor that would be present were the gas phase saturated with water at the specified temperature and pressure. ❷ The value of the logarithmic term of Eq. 13.36 depends on the composition of the environment. In the present case, this term contributes about 3% to the magnitude of the chemical exergy. The contribution of the logarithmic term is usually small. In such instances, a satisfactory approximation to the chemical exergy can be obtained by omitting the term. ✓Skills Developed Ability to... ❑ calculate the chemical exergy of a hydrocarbon fuel relative to a specified reference environment. ❑ calculate the chemical exergy of a hydrocarbon fuel based on standard chemical exergies. Would the higher heating value (HHV) of liquid octane provide a plausible estimate of the chemical exergy in this case? Ans. Yes, Table A-25 gives 47,900 kJ/kg, which is approximately 1% greater than values obtained in parts (a) and (b). Quick Quiz 854 Chapter 13 Reacting Mixtures and Combustion Using Gibbs function data from Table A-25, and standard chemical exergies for O2, N2, and H2O(l) from Table A-26 (Model II), ech NH3 337,910 kJ/kmol. This agrees closely with the value for ammonia listed in Table A-26 for Model II. b b b b b 13.8 Applying Total Exergy The exergy associated with a specified state of a substance is the sum of two contributions: the thermomechanical contribution introduced in Chap. 7 and the chemical contribution introduced in this chapter. On a unit mass basis, the total exergy is e 1u u02 p01y y02 T01s s02 V2 2 gz ech (13.46) where the underlined term is the thermomechanical contribution (Eq. 7.2) and ech is the chemical contribution evaluated as in Sec. 13.6 or 13.7. Similarly, the total flow exergy associated with a specified state is the sum ef h h0 T01s s02 V2 2 gz ech (13.47) where the underlined term is the thermomechanical contribution (Eq. 7.14) and ech is the chemical contribution. 13.8.1 Calculating Total Exergy Exergy evaluations considered in previous chapters of this book have been alike in this respect: Differences in exergy or flow exergy between states of the same composition have been evaluated. In such cases, the chemical exergy contribution cancels, leaving just the difference in thermomechanical contributions to exergy. However, for many evaluations it is necessary to account explicitly for chemical exergy—for instance, chemical exergy is important when evaluating processes involving combustion. When using Eqs. 13.46 and 13.47 to evaluate total exergy at a state, we first think of bringing the substance from that state to the state where it is in thermal and mechanical equilibrium with the environment—that is, to the dead state where temperature is T0 and pressure is p0. In applications dealing with gas mixtures involving water vapor, such as combustion products of hydrocarbons, some condensation of water vapor to liquid may occur in such a process. If so, at the dead state the initial gas mixture consists of a gas phase containing water vapor in equilibrium with liquid water. See the solution of Example 13.15 for an illustration. Yet total exergy evaluations are simplified by assuming at the dead state that all water present in the mixture exists only as a vapor, and this hypothetical dead state condition often suffices. Also see Example 13.15 for an illustration. In Examples 13.13-13.15, we evaluate the total flow exergy for an application involving a cogeneration system.

For computational convenience, the chemical exergy given by Eq. 13.29 is written as Eqs. 13.35 and 13.36. The first of these is obtained by recasting the specific entropies of O2, CO2, and H2O using the following equation obtained by application of Eq. (a) of Table 13.1: si1T0, ye i p02 si1T0, p02 R ln ye i (13.34) The first term on the right is the absolute entropy at T0 and p0, and yi e is the mole fraction of component i in the environment. Applying Eq. 13.34, Eq. 13.29 becomes ech chF aa b 4 c 2 b hO2 ahCO2 b 2 hH2O1g2d 1T0, p02 T0 csF aa b 4 c 2 b sO2 asCO2 b 2 sH2O1g2 d 1T0, p02 (13.35) RT0 ln c 1ye O2 2ab/4c/2 1ye CO2 2a 1ye H2O2b/2 d where the notation (T0, p0) signals that the specific enthalpy and entropy terms of Eq. 13.35 are each evaluated at T0 and p0, although T0 suffices for the enthalpy of substances modeled as ideal gases. Recognizing the Gibbs function in Eq. 13.35—gF hF T0sF , for instance— Eq. 13.35 can be expressed alternatively in terms of the Gibbs functions of the several substances as ech c gF aa b 4 c 2 bgO2 agCO2 b 2 gH2O1g2d 1T0, p02 RT0 ln c 1ye O2 2ab/4c/2 1ye CO2 2a 1ye H2O2b/2 d (13.36) The logarithmic term common to Eqs. 13.35 and 13.36 typically contributes only a few percent to the chemical exergy magnitude. Other observations follow: c The specific Gibbs functions of Eq. 13.36 are evaluated at the temperature T0 and pressure p0 of the environment. These terms can be determined with Eq. 13.28a as g1T0, p02 g f 3g1T0, p02 g1Tref, pref24 (13.37) where gf is the Gibbs function of formation and Tref 5 25C (77F), pref 5 1 atm. c For the special case where T0 and p0 are the same as Tref and pref, respectively, the second term on the right of Eq. 13.37 vanishes and the specific Gibbs function is just the Gibbs function of formation. That is, the Gibbs function values of Eq. 13.36 can be simply read from Tables A-25 or similar compilations. c Finally, note that the underlined term of Eq. 13.36 can be written more compactly as ¢G: the negative of the change in Gibbs function for the reaction, Eq. 13.30, regarding each substance as separate at temperature T0 and pressure p0. 13.6.2 Evaluating Chemical Exergy for Several Cases Cases of practical interest corresponding to selected values of a, b, and c in the representation CaHbOc can be obtained from Eq. 13.36. For example, a 5 8, b 5 18, c 5 0 848 Chapter 13 Reacting Mixtures and Combustion corresponds to octane, C8H18. An application of Eq. 13.36 to evaluate the chemical exergy of octane is provided in Example 13.12. Further special cases follow: c Consider the case of pure carbon monoxide at T0, p0. For CO we have a 5 1, b 5 0, c 5 1. Accordingly, Eq. 13.30 reads CO 1 2 O2 S CO2 , and the chemical exergy obtained from Eq. 13.36 is ech CO 3gCO 1 2 gO2 gCO2 41T0, p02 RT0 ln c 1ye O2 21/2 ye CO2 d (13.38) If carbon monoxide is not pure but a component of an ideal gas mixture at T0, p0, each component i of the mixture enters the control volume of Fig. 13.6 at temperature T0 and the respective partial pressure yip0. The contribution of carbon monoxide to the chemical exergy of the mixture, per mole of CO, is then given by Eq. 13.38, but with the mole fraction of carbon monoxide in the mixture, yCO, appearing in the numerator of the logarithmic term that then reads ln3yCO1ye O2 21/2 /ye CO2 4. This becomes important when evaluating the exergy of combustion products involving carbon monoxide. c Consider the case of pure water at T0 and p0. Water is a liquid when at T0, p0, but is a vapor within the environment of Table 13.4. Thus, water enters the control volume of Fig. 13.6 as a liquid and exits as a vapor at T0, ye H2 Op0 , with no chemical reaction required. In this case, a 5 0, b 5 2, and c 5 1. Equation 13.36 gives the chemical exergy as ech H2O 3gH2O1l2 gH2O1g241T0, p02 RT0 ln a 1 ye H2O b (13.39) c Consider the case of pure carbon dioxide at T0, p0. Like water, carbon dioxide is present within the environment and thus requires no chemical reaction to evaluate its chemical exergy. With a 5 1, b 5 0, c 5 2, Eq. 13.36 gives the chemical exergy simply in terms of a logarithmic expression of the form ech RT0 ln a 1 ye CO2 b (13.40) Provided the appropriate mole fraction ye is used, Eq. 13.40 also applies to other substances that are gases in the environment, in particular to O2 and N2. Moreover, Eqs. 13.39 and 13.40 reveal that a chemical reaction does not always play a part when conceptualizing chemical exergy. In the cases of liquid water, CO2, O2, N2, and other gases present in the environment, we think of the work that could be done as the particular substance passes by diffusion from the dead state, where its pressure is p0, to the environment, where its pressure is the partial pressure, ye p0. c Finally, for an ideal gas mixture at T0, p0 consisting only of substances present as gases in the environment, the chemical exergy is obtained by summing the contributions of each of the components. The result, per mole of mixture, is ech RT0 a j i1 yi ln a yi ye i b (13.41a) where yi and ye i denote, respectively, the mole fraction of component i in the mixture at T0, p0 and in the environment. TAKE NOTE... For liquid water, we think only of the work that could be developed as water expands through a turbine, or comparable device, from pressure p0 to the partial pressure of the water vapor in the environment: H2O(l) at T0, p0 H2O(g) at T0, yH2Op0 e T0 CO2 at T0, p0 CO2 at T0, yCO2 p0 e T0 13.7 Standard Chemical Exergy 849 Expressing the logarithmic term as 1ln11/y i e 2 ln yi2 and introducing a relation like Eq. 13.40 for each gas i, Eq. 13.41a can be written alternatively as ech a j i1 yi ech i RT0 a j i1 yi ln yi (13.41b) The development of Eqs. 13.41a and 13.41b is left as an exercise. 13.6.3 Closing Comments The approach introduced in this section for conceptualizing the chemical exergy of the set of substances represented by CaHbOc can also be applied, in principle, for other substances. In any such application, the chemical exergy is the maximum theoretical work that could be developed by a control volume like that considered in Fig. 13.6 where the substance of interest enters the control volume at T0, p0 and reacts completely with environmental components to produce environmental components. All participating environmental components enter and exit the control volume at their conditions within the environment. By describing the environment appropriately, this approach can be applied to many substances of practical interest.

Since hydrogen is not naturally occurring, it must be produced. Production methods include electrolysis of water (see Sec. 2.7) and chemically reforming hydrogenbearing fuels, predominantly hydrocarbons. See the following box

The fuel cells shown in Fig. 13.3 are proton exchange membrane fuel cells (PEMFCs). At the anode, hydrogen ions (H1) and electrons (e2) are produced. At the cathode, oxygen, hydrogen ions, and electrons react to produce water. c The fuel cell shown schematically in Fig. 13.3a operates with hydrogen (H2) as the fuel and oxygen (O2) as the oxidizer. The reactions at these electrodes and the overall cell reaction are labeled on the figure. The only products of this fuel cell are water, power generated, and waste heat

13.3.3 Closing Comments For a specified fuel and specified temperature and pressure of the reactants, the maximum adiabatic flame temperature is for complete combustion with the theoretical amount of air. The measured value of the temperature of the combustion products may be several hundred degrees below the maximum adiabatic flame temperature, however, for several reasons: c Once adequate oxygen has been provided to permit complete combustion, bringing in more air dilutes the combustion products, lowering their temperature. c Incomplete combustion also tends to reduce the temperature of the products, and combustion is seldom complete (see Sec. 14.4). c Heat losses can be reduced but not altogether eliminated. c As a result of the high temperatures achieved, some of the combustion products may dissociate. Endothermic dissociation reactions lower the product temperature. The effect of dissociation on the adiabatic flame temperature is considered in Sec. 14.4. 13.4 Fuel Cells A fuel cell is an electrochemical device in which fuel and an oxidizer (normally oxygen from air) undergo a chemical reaction, providing electrical current to an external circuit and producing products. The fuel and oxidizer react catalytically in stages on separate electrodes: the anode and the cathode. An electrolyte separating the two electrodes allows passage of ions formed by reaction. Depending on the type of fuel cell, the ions may be positively or negatively charged. Individual fuel cells are connected in parallel or series to form fuel cell stacks to provide the desired level of power output. With today's technology, the preferred fuel for oxidation at the fuel cell anode is hydrogen because of its exceptional ability to produce electrons when suitable catalysts are used, while producing no harmful emissions from the fuel cell itself. Depending on the type of fuel cell, methanol (CH3OH) and carbon monoxide (CO) can be oxidized at the anode in some applications but often with performance penalties.

3.2.3 Enthalpy of Combustion and Heating Values Although the enthalpy of formation concept underlies the formulations of the energy balances for reactive systems presented thus far, the enthalpy of formation of fuels is not always tabulated. fuel oil and coal are normally composed of several individual chemical substances, the relative amounts of which may vary considerably, depending on the source. Owing to the wide variation in composition that these fuels can exhibit, we do not find their enthalpies of formation listed in Tables A-25 or similar compilations of thermophysical data. b b b b b In many cases of practical interest, however, the enthalpy of combustion, which is accessible experimentally, can be used to conduct an energy analysis when enthalpy of formation data are lacking. The enthalpy of combustion hRP is defined as the difference between the enthalpy of the products and the enthalpy of the reactants when complete combustion occurs at a given temperature and pressure. That is, hRP a P nehe a R nihi (13.18) where the n's correspond to the respective coefficients of the reaction equation giving the moles of reactants and products per mole of fuel. When the enthalpy of combustion is expressed on a unit mass of fuel basis, it is designated hRP. Tabulated values are usually given at the standard temperature Tref and pressure pref introduced in Sec. 13.2.1. The symbol h RP or h RP is used for data at this temperature and pressure. The heating value of a fuel is a positive number equal to the magnitude of the enthalpy of combustion. Two heating values are recognized by name: the higher heating value (HHV) and the lower heating value (LHV). The higher heating value is obtained when all the water formed by combustion is a liquid; the lower heating value

A fuel is simply a combustible substance. In this chapter emphasis is on hydrocarbon fuels, which contain hydrogen and carbon. Sulfur and other chemical substances also may be present. Hydrocarbon fuels can exist as liquids, gases, and solids. Liquid hydrocarbon fuels are commonly derived from crude oil through distillation and cracking processes. Examples are gasoline, diesel fuel, kerosene, and other types of fuel oils. Most liquid fuels are mixtures of hydrocarbons for which compositions are usually given in terms of mass fractions. For simplicity in combustion calculations, gasoline is often modeled as octane, C8H18, and diesel fuel as dodecane, C12H26. Gaseous hydrocarbon fuels are obtained from natural gas wells or are produced in certain chemical processes. Natural gas normally consists of several different hydrocarbons, with the major constituent being methane, CH4. The compositions of gaseous fuels are usually given in terms of mole fractions. Both gaseous and liquid hydrocarbon fuels can be synthesized from coal, oil shale, and tar sands. Coal is a familiar solid fuel. Its composition varies considerably with the location from which it is mined. For combustion calculations, the composition of coal is usually expressed as an ultimate analysis. The ultimate analysis gives the composition on a mass basis in terms of the relative amounts of chemical elements (carbon, sulfur, hydrogen, nitrogen, oxygen) and ash. 13.1.2 Modeling Combustion Air Oxygen is required in every combustion reaction. Pure oxygen is used only in special applications such as cutting and welding. In most combustion applications, air provides the needed oxygen. The composition of a typical sample of dry air is given in Table 12.1. For the combustion calculations of this book, however, the following model is used for simplicity: c All components of dry air other than oxygen are lumped together with nitrogen. Accordingly, air is considered to be 21% oxygen and 79% nitrogen on a molar basis. With this idealization the molar ratio of the nitrogen to the oxygen is 0.79/0.21 5 3.76. When air supplies the oxygen in a combustion reaction, therefore, every mole of oxygen is accompanied by 3.76 moles of nitrogen. c We also assume that the nitrogen present in the combustion air does not undergo chemical reaction. That is, nitrogen is regarded as inert. The nitrogen in the products is at the same temperature as the other products, however. Accordingly, nitrogen undergoes a change of state if the products are at a temperature other than the reactant air temperature. If a high enough product temperature is attained, nitrogen can form compounds such as nitric oxide and nitrogen dioxide. Even trace amounts of oxides of nitrogen appearing in the exhaust of internal combustion engines can be a source of air pollution. Air-Fuel Ratio Two parameters that are frequently used to quantify the amounts of fuel and air in a particular combustion process are the air-fuel ratio and its reciprocal, the fuel-air ratio. The air-fuel ratio is simply the ratio of the amount of air in a reaction to the amount of fuel. The ratio can be written on a molar basis (moles of air divided by moles of fuel) or on a mass basis (mass of air divided by mass of fuel). Conversion between these values is accomplished using the molecular weights of the air, Mair, and fuel, Mfuel,

CLOSED SYSTEMS. Next consider an entropy balance for a process of a closed system during which a chemical reaction occurs SP SR a dQ T b b s (13.25) SR and SP denote, respectively, the entropy of the reactants and the entropy of the products. When the reactants and products form ideal gas mixtures, the entropy balance can be expressed on a per mole of fuel basis as a P ns a R ns 1 nF a dQ T b b s nF (13.26) where the coefficients n on the left are the coefficients of the reaction equation giving the moles of each reactant or product per mole of fuel. The entropy terms of Eq. 13.26 are evaluated from Eq. 13.23 using the temperature and partial pressures of the reactants or products, as appropriate. In any such application, the fuel is mixed with the oxidizer, so this must be taken into account when determining the partial pressures of the reactants. Example 13.10 provides an illustration of the evaluation of entropy change for combustion at constant volume.

Devices designed to do work by utilization of a combustion process, such as vapor and gas power plants and reciprocating internal combustion engines, invariably have irreversibilities and losses associated with their operation. Accordingly, actual devices produce work equal to only a fraction of the maximum theoretical value that might be obtained. The vapor power plant exergy analysis of Sec. 8.6 and the combined cycle exergy analysis of Example 9.12 provide illustrations. The performance of devices whose primary function is to do work can be evaluated as the ratio of the actual work developed to the exergy of the fuel consumed in producing that work. This ratio is an exergetic efficiency. The relatively low exergetic efficiency exhibited by many common power-producing devices suggests that thermodynamically more thrifty ways of utilizing the fuel to develop power might be possible. However, efforts in this direction must be tempered by the economic imperatives that govern the practical application of all devices. The trade-off between fuel savings and the additional costs required to achieve those savings must be carefully weighed. The fuel cell provides an illustration of a relatively fuel-efficient device. We noted previously (Sec. 13.4) that the chemical reactions in fuel cells are more controlled than the rapidly occurring, highly irreversible combustion reactions taking place in conventional power-producing systems. In principle, fuel cells can achieve greater exergetic efficiencies than many such devices. Still, relative to conventional power systems, fuel cell systems typically cost more per unit of power generated, and this has limited their deployment.

Evaluating Enthalpy of Combustion by Calorimetry When enthalpy of formation data are available for all the reactants and products, the enthalpy of combustion can be calculated directly from Eq. 13.18, as illustrated in Example 13.7. Otherwise, it must be obtained experimentally using devices known as calorimeters. Both constant-volume (bomb calorimeters) and flow-through devices are employed for this purpose. Consider as an illustration a reactor operating at steady state in which the fuel is burned completely with air. For the products to be returned to the same temperature as the reactants, a heat transfer from the reactor would be required. From an energy rate balance, the required heat transfer is Qcv # n # F a P nehe a R ni hi (13.19) where the symbols have the same significance as in previous discussions. The heat transfer per mole of fuel, Q # cvn # F, would be determined from measured data. Comparing Eq. 13.19 with the defining equation, Eq. 13.18, we have hRP Q # cvn # F. In accord with the usual sign convention for heat transfer, the enthalpy of combustion would be negative. As noted previously, the enthalpy of combustion can be used for energy analyses of reacting systems. consider a control volume at steady state in which a fuel oil reacts completely with air. The energy rate balance is given by Eq. 13.15b Q # cv n # F W # cv n # F a P ne1hf ¢h2e a R ni1hf ¢h2i All symbols have the same significance as in previous discussions. This equation can be rearranged to read Q # cv n # F W # cv n # F a P ne1hf2e a R ni1hf2i a P ne1¢h2e a R ni1¢h2i For a complete reaction, the underlined term is just the enthalpy of combustion h RP, at Tref and pref. Thus, the equation becomes Q # cv n # F W # cv n # F h RP a P ne1¢h2e a R ni1¢h2i (13.20) The right side of Eq. 13.20 can be evaluated with an experimentally determined value for h RP and ¢h values for the reactants and products determined as discussed previously. b b b b b 13.3 Determining the Adiabatic Flame Temperature Let us reconsider the reactor at steady state pictured in Fig. 13.2. In the absence of work W # cv and appreciable kinetic and potential energy effects, the energy liberated on combustion is transferred from the reactor in two ways only: by energy accompanying the exiting combustion products and by heat transfer to the surroundings. The smaller the heat transfer, the greater the energy carried out with the combustion products and thus the greater the temperature of the products. The temperature that would be achieved by the products in the limit of adiabatic operation of the reactor is called the adiabatic flame temperature or adiabatic combustion temperature. The adiabatic flame temperature can be determined by use of the conservation of mass and conservation of energy principles. To illustrate the procedure, let us suppose that the combustion air and the combustion products each form ideal gas mixtures.

Hydrocarbon reforming can occur either separately or within the fuel cell (depending on type). Hydrogen produced by reforming fuel separately from the fuel cell itself is known as external reforming. If not fed directly from the reformer to a fuel cell, hydrogen can be stored as a compressed gas, a cryogenic liquid, or atoms absorbed within metallic structures and then provided to fuel cells from storage, when required. Internal reforming refers to applications where hydrogen production by reforming fuel is integrated within the fuel cell. Owing to limitations of current technology, internal reforming is feasible only in fuel cells operating at temperatures above about 600C. Rates of reaction in fuel cells are limited by the time it takes for diffusion of chemical species through the electrodes and the electrolyte and by the speed of the chemical reactions themselves. The reaction in a fuel cell is not a combustion process. These features result in fuel cell internal irreversibilities that are inherently less significant than those encountered in power systems employing combustion. Thus, fuel cells have the potential of providing more power from a given supply of fuel and oxidizer than conventional internal combustion engines and gas turbines. Fuel cells do not operate as thermodynamic power cycles, and thus the notion of a limiting thermal efficiency imposed by the second law is not applicable. However, as for all power systems, the power provided by fuel cell systems is eroded by inefficiencies in auxiliary equipment. For fuel cells this includes heat exchangers, compressors, and humidifiers. Irreversibilities and losses inherent in hydrogen production also can be greater than those seen in production of more conventional fuels. In comparison to reciprocating internal combustion engines and gas turbines that incorporate combustion, fuel cells typically produce relatively few damaging emissions as they develop power. Still, such emissions accompany production of fuels consumed by fuel cells as well as the manufacture of fuel cells and their supporting components. Despite potential thermodynamic advantages, widespread fuel cell use has not occurred thus far owing primarily to cost. Table 13.2 summarizes the most promising fuel cell technologies currently under consideration. Included are potential applications and other characteristics. Cooperative efforts by government and industry have fostered advances in both proton exchange membrane fuel cells and solid oxide fuel cells, which appear to provide the greatest range of potential applications in transportation, portable power, and stationary power. The proton exchange membrane fuel cell and the solid oxide fuel cell are discussed next.

In each of the illustrations given above, complete combustion is assumed. For a hydrocarbon fuel, this means that the only allowed products are CO2, H2O, and N2, with O2 also present when excess air is supplied. If the fuel is specified and combustion is complete, the respective amounts of the products can be determined by applying the conservation of mass principle to the chemical equation. The procedure for obtaining the balanced reaction equation of an actual reaction where combustion is incomplete is not always so straightforward. Combustion is the result of a series of very complicated and rapid chemical reactions, and the products formed depend on many factors. When fuel is burned in the cylinder of an internal combustion engine, the products of the reaction vary with the temperature and pressure in the cylinder. In combustion equipment of all kinds, the degree of mixing of the fuel and air is a controlling factor in the reactions that occur once the fuel and air mixture is ignited. Although the amount of air supplied in an actual combustion process may exceed the theoretical amount, it is not uncommon for some carbon monoxide and unburned oxygen to appear in the products. This can be due to incomplete mixing, insufficient time for complete combustion, and other factors. When the amount of air supplied is less than the theoretical amount of air, the products may include both CO2 and CO, and there also may be unburned fuel in the products. Unlike the complete combustion cases considered above, the products

In this section, we consider a thought experiment to bring out important aspects of chemical exergy. This involves c a set of substances represented by CaHbOc (see Table 13.3), c an environment modeling Earth's atmosphere (see Table 13.4), and c an overall system including a control volume (see Fig. 13.6). Referring to Table 13.4, the exergy reference environment considered in the present discussion is an ideal gas mixture modeling Earth's atmosphere. T0 and p0 denote the temperature and pressure of the environment, respectively. The composition of the environment is given in terms of mole fractions denoted by ye , where superscript e is used to signal the mole fraction of an environmental component. The values of these mole fractions, and the values of T0 and p0, are specified and remain unchanged throughout the development to follow. The gas mixture modeling the atmosphere adheres to the Dalton model (Sec. 12.2). Considering Fig. 13.6, a substance represented by CaHbOc enters the control volume at T0, p0. Depending on the particular substance, compounds present in the environment enter (O2) and exit (CO2 and H2O(g)) at T0 and their respective partial pressures in the environment. All substances enter and exit with negligible effects of motion and gravity. Heat transfer between the control volume and environment occurs only at temperature T0. The control volume operates at steady state, and the ideal gas model applies to all gases. Finally, for the overall system whose boundary is denoted by the dotted line, total volume is constant and there is no heat transfer across the boundary. Next, we apply conservation of mass, an energy balance, and an entropy balance to the control volume of Fig. 13.6 with the objective of determining the maximum theoretical work per mole of substance CaHbOc entering—namely, the maximum theoretical value of W # cv /n # F. This value is the molar chemical exergy of the substance. The chemical exergy is given by e ch chF aa b 4 c 2 b hO2 ahCO2 b 2 hH2O d T0 csF aa b 4 c 2 b sO2 asCO2 b 2 sH2Od (13.29) where the superscript ch is used to distinguish this contribution to the exergy magnitude from the thermomechanical exergy introduced in Chap. 7. The subscript F Fig. 13.6 Illustration used to conceptualize chemical exergy. Boundary of overall system Boundary of the control volume Heat transfer with environment Environment at T0, p0 CaHbOc at T0, p0 T0 CO2 at T0, yCO2 p0 e CaHbOc + [a + b/4 − c/2]O2 → aCO2 + b/2 H2O(g) O2 at T0, yO2 p0 e H2O at T0, yH2Op0 e Wcv/nF Set of Substances Represented by CaHbOc C H2 CaHb CO CO2 H2O(liq.) a 1 0 a 1 1 0 b 0 2 b 0 0 2 C 0 0 0 1 2 1 TABLE 13.3 Exergy Reference Environment Used in Sec. 13.6 Gas phase at T0 298.15 K (25C), p0 1 atm Component ye (%) N2 75.67 O2 20.35 H2O(g) 3.12 CO2 0.03 Other 0.83 TABLE 13.4 846 Chapter 13 Reacting Mixtures and Combustion denotes the substance represented by CaHbOc. The other molar enthalpies and entropies appearing in Eq. 13.29 refer to the substances entering and exiting the control volume, each evaluated at the state at which it enters or exits. See the following box for the derivation of Eq. 13.29.

Normally the amount of air supplied is either greater or less than the theoretical amount. The amount of air actually supplied is commonly expressed in terms of the percent of theoretical air. For example, 150% of theoretical air means that the air actually supplied is 1.5 times the theoretical amount of air. The amount of air supplied can be expressed alternatively as a percent excess air or a percent deficiency of air. Thus, 150% of theoretical air is equivalent to 50% excess air, and 80% of theoretical air is the same as a 20% deficiency of air

The discussion continues in the second part of this chapter dealing with the exergy concept and in the next chapter where the subject of chemical equilibrium is taken up. 13.5.1 Evaluating Entropy for Reacting Systems The property entropy plays an important part in quantitative evaluations using the second law of thermodynamics. When reacting systems are under consideration, the same problem arises for entropy as for enthalpy and internal energy: A common datum must be used to assign entropy values for each substance involved in the reaction. This is accomplished using the third law of thermodynamics and the absolute entropy concept. The third law deals with the entropy of substances at the absolute zero of temperature. Based on empirical evidence, this law states that the entropy of a pure crystalline substance is zero at the absolute zero of temperature, 0 K or 0R. Substances not having a pure crystalline structure at absolute zero have a nonzero value of entropy at absolute zero. The experimental evidence on which the third law is based is obtained primarily from studies of chemical reactions at low temperatures and specific heat measurements at temperatures approaching absolute zero. ABSOLUTE ENTROPY. For present considerations, the importance of the third law is that it provides a datum relative to which the entropy of each substance participating in a reaction can be evaluated so that no ambiguities or conflicts arise. The entropy relative to this datum is called the absolute entropy. The change in entropy of a substance between absolute zero and any given state can be determined from precise measurements of energy transfers and specific heat data or from procedures based on statistical thermodynamics and observed molecular data. Tables A-25 and A-25E give the value of the absolute entropy for selected substances at the standard reference state, Tref 5 298.15 K, pref 5 1 atm, in units of kJ/ kmol ? K and Btu/lbmol ? 8R, respectively. Two values of absolute entropy for water are provided. One is for liquid water and the other is for water vapor. As for the case of the enthalpy of formation of water considered in Sec. 13.2.1, the vapor value listed is for a hypothetical ideal gas state in which water is a vapor at 258C (778F) and pref 5 1 atm. Tables A-23 and A-23E give tabulations of absolute entropy versus temperature at a pressure of 1 atm for selected gases. In these tables, the absolute entropy at 1 atm and temperature T is designated s1T2, and ideal gas behavior is assumed for the gases. USING ABSOLUTE ENTROPY. When the absolute entropy is known at the standard state, the specific entropy at any other state can be found by adding the specific entropy change between the two states to the absolute entropy at the standard state. Similarly, when the absolute entropy is known at the pressure pref and temperature T, the absolute entropy at the same temperature and any pressure p can be found from s1T, p2 s1T, pref2 3s1T, p2 s1T, pref24 For the ideal gases listed in Tables A-23, the first term on the right side of this equation is s1T2, and the second term on the right can be evaluated using Eq. 6.18. Collecting results, we get s1T, p2 s1T2 R ln p pref (ideal gas) (13.22) To reiterate, s1T2 is the absolute entropy at temperature T and pressure pref 1 atm. The entropy of the ith component of an ideal gas mixture is evaluated at the mixture temperature T and the partial pressure pi : si1T, pi2. The partial pressure is given

The objective of the present section is to illustrate the application of the conservation of energy principle to reacting systems. The forms of the conservation of energy principle introduced previously remain valid whether or not a chemical reaction occurs within the system. However, the methods used for evaluating the properties of reacting systems differ somewhat from the practices used to this point. 13.2.1 Evaluating Enthalpy for Reacting Systems In most tables of thermodynamic properties used thus far, values for the specific internal energy, enthalpy, and entropy are given relative to some arbitrary datum state where the enthalpy (or alternatively the internal energy) and entropy are set to zero. This approach is satisfactory for evaluations involving differences in property values between states of the same composition, for then arbitrary datums cancel. However, when a chemical reaction occurs, reactants disappear and products are formed, so differences cannot be calculated for all substances involved. For reacting systems, it is necessary to evaluate h, u, and s in such a way that there are no subsequent ambiguities or inconsistencies in evaluating properties. In this section, we will consider

The objective of this part of the chapter is to extend the exergy concept introduced in Chap. 7 to include chemical exergy. Several important exergy aspects are listed in Sec. 7.3.1. We suggest you review that material before continuing the current discussion. A key aspect of the Chap. 7 presentation is that thermomechanical exergy is a measure of the departure of the temperature and pressure of a system from those of a thermodynamic model of Earth and its atmosphere called the exergy reference environment or, simply, the environment. In the current discussion, the departure of the system state from the environment centers on composition, for chemical exergy is a measure of the departure of the composition of a system from that of the exergy reference environment. Exergy is the maximum theoretical work obtainable from an overall system of system plus environment as the system passes from a specified state to equilibrium with the environment. Alternatively, exergy is the minimum theoretical work input required to form the system from the environment and bring it to the specified state. For conceptual and computational ease, we think of the system passing to equilibrium with the environment in two steps. With this approach, exergy is the sum of two contributions: thermomechanical, developed in Chap. 7, and chemical, developed in this chapter

The state of the combustion products also must be assessed. For instance, the presence of water vapor should be noted, for some of the water present will condense if the products are cooled sufficiently. The energy balance must then be written to account for the presence of water in the products as both a liquid and a vapor. For cooling of combustion products at constant pressure, the dew point temperature method of Example 13.2 is used to determine the temperature at the onset of condensation. Analyzing Control Volumes at Steady State To illustrate the many considerations involved when writing energy balances for reacting systems, we consider special cases of broad interest, highlighting the underlying assumptions. Let us begin by considering the steady-state reactor shown in Fig. 13.2, in which a hydrocarbon fuel CaHb burns completely with the theoretical amount of air according to CaHb aa b 4 b1O2 3.76N22 S aCO2 b 2 H2O aa b 4 b 3.76N2 (13.11) The fuel enters the reactor in a stream separate from the combustion air, which is regarded as an ideal gas mixture. The products of combustion also are assumed to form an ideal gas mixture. Kinetic and potential energy effects are ignored. With the foregoing idealizations, the mass and energy rate balances for the twoinlet, single-exit reactor can be used to obtain the following equation on a per mole of fuel basis: Q # cv n # F W # cv n # F c ahCO2 b 2 hH2O aa b 4 b 3.76 hN2 d hF c aa b 4 b hO2 aa b 4 b 3.76 hN2 d (13.12a) where n # F denotes the molar flow rate of the fuel. Note that each coefficient on the right side of this equation is the same as the coefficient of the corresponding substance in the reaction equation. The first underlined term on the right side of Eq. 13.12a is the enthalpy of the exiting gaseous products of combustion per mole of fuel. The second underlined term on the right side is the enthalpy of the combustion air per mole of fuel. In accord with Table 13.1, the enthalpies of the combustion products and the air have been evaluated by adding the contribution of each component present in the respective ideal gas mixtures. The symbol hF denotes the molar enthalpy of the fuel. Equation 13.12a can be expressed more concisely as Q # cv n # F W # cv n # F hP hR (13.12b) where hP and hR denote, respectively, the enthalpies of the products and reactants per mole of fuel. EVALUATING ENTHALPY TERMS. Once the energy balance has been written, the next step is to evaluate the individual enthalpy terms. Since each component of

The thermodynamic property known as the Gibbs function plays a role in the second part of this chapter dealing with exergy analysis. The specific Gibbs function g, introduced in Sec. 11.3, is g h T s (13.27) The procedure followed in setting a datum for the Gibbs function closely parallels that used in defining the enthalpy of formation: To each stable element at the standard state is assigned a zero value of the Gibbs function. The Gibbs function of formation of a compound, g f , equals the change in the Gibbs function for the reaction in which the compound is formed from its elements, the compound and the elements all being at Tref 5 258C (778F) and pref 5 1 atm. Tables A-25 and A-25E give the Gibbs function of formation, g f , for selected substances. The Gibbs function at a state other than the standard state is found by adding to the Gibbs function of formation the change in the specific Gibbs function ¢g between the standard state and the state of interest: g1T, p2 g f 3g1T, p2 g1Tref, pref24 gf ¢g (13.28a) With Eq. 13.27, ¢g can be written as ¢g 3h1T, p2 h1Tref, pref24 3T s 1T, p2 Tref s1Tref, pref24 (13.28b) The Gibbs function of component i in an ideal gas mixture is evaluated at the partial pressure of component i and the mixture temperature. The procedure for determining the Gibbs function of formation is illustrated in the next example.

Then, with the other assumptions stated previously, the energy rate balance on a per mole of fuel basis, Eq. 13.12b, reduces to the form hP hR—that is, a P nehe a R nihi (13.21a) where i denotes the incoming fuel and air streams and e the exiting combustion products. With this expression, the adiabatic flame temperature can be determined using table data or computer software, as follows. 13.3.1 Using Table Data When using Eq. 13.9 with table data to evaluate enthalpy terms, Eq. 13.21a takes the form a P ne1hf ¢h2e a R ni1hf ¢h2i or a P ne1¢h2e a R ni1¢h2i a R nihfi a P nehfe (13.21b) The n's are obtained on a per mole of fuel basis from the balanced chemical reaction equation. The enthalpies of formation of the reactants and products are obtained from Table A-25 or A-25E. Enthalpy of combustion data might be employed in situations where the enthalpy of formation for the fuel is not available. Knowing the states of the reactants as they enter the reactor, the ¢h terms for the reactants can be evaluated as discussed previously. Thus, all terms on the right side of Eq. 13.21b can be evaluated. The terms (¢h)e on the left side account for the changes in enthalpy of the products from Tref to the unknown adiabatic flame temperature. Since the unknown temperature appears in each term of the sum on the left side of the equation, determination of the adiabatic flame temperature requires iteration: A temperature for the products is assumed and used to evaluate the left side of Eq. 13.21b. The value obtained is compared with the previously determined value for the right side of the equation. The procedure continues until satisfactory agreement is attained. Example 13.8 gives an illustration. 13.3.2 Using Computer Software Thus far we have emphasized the use of Eq. 13.9 together with table data when evaluating the specific enthalpies required by energy balances for reacting systems. Such enthalpy values also can be retrieved using Interactive Thermodynamics: IT. With IT, the quantities on the right side of Eq. 13.9 are evaluated by software, and h data are returned directly. consider CO2 at 500 K modeled as an ideal gas. The specific enthalpy is obtained from IT as follows: T = 500 // K h = h_T("CO2", T) Choosing K for the temperature unit and moles for the amount under the Units menu, IT returns h 5 23.852 3 105 kJ/kmol. This value agrees with the value calculated from Eq. 13.9 using enthalpy data for CO2 from Table A-23, as follows h hf 3h1500 K2 h1298 K24 393,520 317,678 93644 3.852 105 kJ/kmol b b b b b

Thus far our study of reacting systems has involved only the conservation of mass principle. A more complete understanding of reacting systems requires application of the first and second laws of thermodynamics. For these applications, energy and entropy balances play important roles, respectively. Energy balances for reacting systems are developed and applied in Secs. 13.2 and 13.3; entropy balances for reacting systems are the subject of Sec. 13.5. To apply such balances, it is necessary to take special care in how internal energy, enthalpy, and entropy are evaluated. For the energy and entropy balances of this chapter, combustion air and (normally) products of combustion are modeled as ideal gas mixtures. Accordingly, ideal gas mixture principles introduced in the first part of Chap. 12 play a role. For ease of reference, Table 13.1 summarizes ideal gas mixture relations introduced in Chap. 12 that are used in this chapter.

When a chemical reaction occurs, the bonds within molecules of the reactants are broken, and atoms and electrons rearrange to form products. In combustion reactions, rapid oxidation of combustible elements of the fuel results in energy release as combustion products are formed. The three major combustible chemical elements in most common fuels are carbon, hydrogen, and sulfur. Sulfur is usually a relatively unimportant contributor to the energy released, but it can be a significant cause of pollution and corrosion problems. Combustion is complete when all the carbon present in the fuel is burned to carbon dioxide, all the hydrogen is burned to water, all the sulfur is burned to sulfur dioxide, and all other combustible elements are fully oxidized. When these conditions are not fulfilled, combustion is incomplete. In this chapter, we deal with combustion reactions expressed by chemical equations of the form reactants S products or fuel oxidizer S products When dealing with chemical reactions, it is necessary to remember that mass is conserved, so the mass of the products equals the mass of the reactants. The total mass of each chemical element must be the same on both sides of the equation, even though the elements exist in different chemical compounds in the reactants and products. However, the number of moles of products may differ from the number of moles of reactants.

While the approach used in Sec. 13.6 for conceptualizing chemical exergy can be applied to many substances of practical interest, complications are soon encountered. For one thing, the environment generally must be extended; the simple environment of Table 13.4 no longer suffices. In applications involving coal, for example, sulfur dioxide or some other sulfur-bearing compound must appear among the environmental components. Furthermore, once the environment is determined, a series of calculations is required to obtain exergy values for the substances of interest. These complexities can be sidestepped by using a table of standard chemical exergies. Standard chemical exergy values are based on a standard exergy reference environment exhibiting standard values of the environmental temperature T0 and pressure p0 such as 298.15 K (536.678R) and 1 atm, respectively. The exergy reference environment also consists of a set of reference substances with standard concentrations reflecting as closely as possible the chemical makeup of the natural environment. To exclude the possibility of developing work from interactions among parts of the environment, these reference substances must be in equilibrium mutually. The reference substances generally fall into three groups: gaseous components of the atmosphere, solid substances from Earth's crust, and ionic and nonionic substances from the oceans. A common feature of standard exergy reference environments is a gas phase, intended to represent air, that includes N2, O2, CO2, H2O(g), and other gases. The ith gas present in this gas phase is assumed to be at temperature T0 and the partial pressure pe i ye i p0. Two standard exergy reference environments are considered in this book, called Model I and Model II. For each of these models, Table A-26 gives values of the standard chemical exergy for several substances, in units of kJ/kmol, together with a brief description of the underlying rationale. The methods employed to determine the tabulated standard chemical exergy values are detailed in the references accompanying the tables. Only one of the two models should be used in a particular analysis. The use of a table of standard chemical exergies often simplifies the application of exergy principles. However, the term standard is somewhat misleading, for there is no one specification of the environment that suffices for all applications. Still, chemical exergies calculated relative to alternative specifications of the environment are generally in good agreement. For a broad range of engineering applications, the

by pi yip, where yi is the mole fraction of component i and p is the mixture pressure. Thus, Eq. 13.22 takes the form si1T, pi2 si1T2 R ln pi pref or si1T, pi2 si1T2 R ln yip pref a component i of an ideal gas mixture b (13.23) where si1T2 is the absolute entropy of component i at temperature T and pref 1 atm. Equation 13.23 corresponds to Eq. (b) of Table 13.1. Finally, note that Interactive Thermodynamics (IT) returns absolute entropy directly and does not use the special function s. 13.5.2 Entropy Balances for Reacting Systems Many of the considerations that enter when energy balances are written for reacting systems also apply to entropy balances. The writing of entropy balances for reacting systems will be illustrated by referring to special cases of broad interest. CONTROL VOLUMES AT STEADY STATE. Let us begin by reconsidering the steady-state reactor of Fig. 13.2, for which the combustion reaction is given by Eq. 13.11. The combustion air and the products of combustion are each assumed to form ideal gas mixtures, and thus Eq. 12.26 from Table 13.1 for mixture entropy is applicable to them. The entropy rate balance for the two-inlet, single-exit reactor can be expressed on a per mole of fuel basis as 0 a i Q # j /Tj n # F sF c aa b 4 bsO2 aa b 4 b 3.76sN2 d c asCO2 b 2 sH2O aa b 4 b 3.76sN2 d s # cv n # F (13.24) where n # F is the molar flow rate of the fuel and the coefficients appearing in the underlined terms are the same as those for the corresponding substances in the reaction equation. All entropy terms of Eq. 13.24 are absolute entropies. The first underlined term on the right side of Eq. 13.24 is the entropy of the combustion air per mole of fuel. The second underlined term is the entropy of the exiting combustion products per mole of fuel. In accord with Table 13.1, the entropies of the air and combustion products are evaluated by adding the contribution of each component present in the respective gas mixtures. For instance, the specific entropy of a substance in the combustion products is evaluated from Eq. 13.23 using the temperature of the combustion products and the partial pressure of the substance in the combustion product mixture. Such considerations are illustrated in Example 13.9.

c The fuel cell shown schematically in Fig. 13.3b operates with humidified methanol (CH3OH 1 H2O) as the fuel and oxygen (O2) as the oxidizer. This type of PEMFC is a direct-methanol fuel cell. The reactions at these electrodes and the overall cell reaction are labeled on the figure. The only products of this fuel cell are water, carbon dioxide, power generated, and waste heat. For PEMFCs, charge-carrying hydrogen ions are conducted through the electrolytic membrane. For acceptable ion conductivity, high membrane water content is required. This requirement restricts the fuel cell to operating below the boiling point of water, so PEMFCs typically operate at temperatures in the range 60-80C. Cooling is generally needed to maintain the fuel cell at the operating temperature. Owing to the relatively low-temperature operation of proton exchange membrane fuel cells, hydrogen derived from hydrocarbon feedstock must be produced using external reforming, while costly platinum catalysts are required at both the anode and cathode to increase ionization reaction rates. Due to an extremely slow reaction rate at the anode, the direct-methanol fuel cell requires several times as much platinum catalyst as the hydrogen-fueled PEMFC to improve the anode reaction rate. Catalytic activity is more important in lower-temperature fuel cells because rates of reaction at the anode and cathode tend to decrease with decreasing temperature. Automakers continue to strive toward the production of market-ready hydrogenfueled proton exchange membrane fuel cell vehicles. Still, just two such fuel cell vehicles are currently available in the United States. To promote further fuel cell

consider the complete combustion of methane with 150% theoretical air (50% excess air). The balanced chemical reaction equation is CH4 11.521221O2 3.76N22 S CO2 2H2O O2 11.28N2 (13.5) In this equation, the amount of air per mole of fuel is 1.5 times the theoretical amount determined by Eq. 13.4. Accordingly, the air-fuel ratio is 1.5 times the air-fuel ratio determined for Eq. 13.4. Since complete combustion is assumed, the products contain only carbon dioxide, water, nitrogen, and oxygen. The excess air supplied appears in the products as uncombined oxygen and a greater amount of nitrogen than in Eq. 13.4, based on the theoretical amount of air. b b b b b The equivalence ratio is the ratio of the actual fuel-air ratio to the fuel-air ratio for complete combustion with the theoretical amount of air. The reactants are said to form a lean mixture when the equivalence ratio is less than unity. When the ratio is greater than unity, the reactants are said to form a rich mixture. In Example 13.1, we use conservation of mass to obtain balanced chemical reactions. The air-fuel ratio for each of the reactions is also calculated.

convenience of using standard values generally outweighs the slight lack of accuracy that might result. In particular, the effect of slight variations in the values of T0 and p0 about their standard values normally can be neglected. 13.7.1 Standard Chemical Exergy of a Hydrocarbon: CaHb In principle, the standard chemical exergy of a substance not present in the environment can be evaluated by considering a reaction of the substance with other substances for which the chemical exergies are known. To illustrate this for the case of a pure hydrocarbon fuel CaHb at T0, p0, refer to the control volume at steady state shown in Fig. 13.7 where the fuel reacts completely with oxygen to form carbon dioxide and liquid water. All substances are assumed to enter and exit at T0, p0 and heat transfer occurs only at temperature T0. Assuming no irreversibilities, an exergy rate balance for the control volume reads 0 a j c1 T0 Tj d 0 a Q # j n # F b a W # cv n # F b int rev e F ch aa b 4 bech O2 aech CO2 a b 2 bech H2O1l2 E # 0 d where the subscript F denotes CaHb. Solving for the chemical exergy ech F , we get ech F a W # cv n # F b int rev aech CO2 a b 2 b ech H2O1l2 aa b 4 b e ch O2 (13.42) Applying energy and entropy balances to the control volume, as in the development in the preceding box giving the derivation of Eq. 13.29, we get a W # cv n # F b int rev c hF aa b 4 b hO2 ahCO2 b 2 hH2O1l2 d 1T0, p02 T0 csF aa b 4 b sO2 asCO2 b 2 sH2O1l2d 1T0, p02 (13.43) The underlined term in Eq. 13.43 is recognized from Sec. 13.2.3 as the molar higher heating value HHV (T0, p0). Substituting Eq. 13.43 into Eq. 13.42, we obtain ech F HHV1T0, p02 T0 csF aa b 4 b sO2 asCO2 b 2 sH2O1l2d 1T0, p02 aech CO2 a b 2 b ech H2O1l2 aa b 4 b ech O2 (13.44a) CaHb at T0, p0 O2 at T0, p0 CO2 at T0, p0 H2O (l) at T0, p0 T0 Qcv · Wcv · CaHb + (a + ) b 4 O2 → aCO2 + H2O(l) b 2 Fig. 13.7 Reactor used to introduce the standard chemical exergy of CaHb. 13.7 Standard Chemical Exergy 851 Equations 13.42 and 13.43 can be expressed alternatively in terms of molar Gibbs functions as follows ech F cgF aa b 4 b gO2 agCO2 b 2 gH2O1l2d 1T0, p02 aech CO2 a b 2 b ech H2O1l2 aa b 4 b ech O2 (13.44b) With Eqs. 13.44, the standard chemical exergy of the hydrocarbon CaHb can be calculated using the standard chemical exergies of O2, CO2, and H2O(l), together with selected property data: the higher heating value and absolute entropies, or Gibbs functions. consider the case of methane, CH4, and T0 5 298.15 K (258C), p0 5 1 atm. For this application we can use Gibbs function data directly from Table A-25, and standard chemical exergies for CO2, H2O(l), and O2 from Table A-26 (Model II), since each source corresponds to 298.15 K, 1 atm. With a 5 1, b 5 4, Eq. 13.44b gives 831,680 kJ/kmol. This agrees with the value listed for methane in Table A-26 for Model II. b b b b b We conclude the present discussion by noting special aspects of Eqs. 13.44: c First, Eq. 13.44a requires the higher heating value and the absolute entropy sF. When data from property compilations are lacking for these quantities, as in the cases of coal, char, and fuel oil, the approach of Eq. 13.44a can be invoked using a measured or estimated heating value and an estimated value of the absolute entropy sF determined with procedures discussed in the literature.2 c Next, note that the first term of Eq. 13.44b can be written more compactly as DG: the negative of the change in Gibbs function for the reaction. c Finally, note that only the underlined terms of Eqs. 13.44 require chemical exergy data relative to the model selected for the exergy reference environment. In Example 13.12 we compare the use of Eq. 13.36 and Eq. 13.44b for evaluating the chemical exergy of a pure hydrocarbon fuel.

development major automakers have formed partnerships among themselves. Thus far, several major brands have formed three distinct partnerships, each having two or three automakers as members. Both hydrogen-fueled and direct-methanol PEMFCs have potential to replace batteries in portable devices such as cellular phones, laptop computers, and video players. Hurdles to wider deployment of PEMFCs include extending stack life, simplifying system integration, and reducing costs. 13.4.2 Solid Oxide Fuel Cell For scale, Fig. 13.4a shows a solid oxide fuel cell (SOFC) module. The fuel cell schematic shown in Fig. 13.4b operates with hydrogen (H2) as the fuel and oxygen (O2) as the oxidizer. At the anode, water (H2O) and electrons (e2) are produced. At the cathode, oxygen reacts with electrons (e2) to produce oxygen ions (O) that migrate through the electrolyte to the anode. The reactions at these electrodes and the overall cell reaction are labeled on the figure. The only products of this fuel cell are water, power generated, and waste heat. For SOFCs, an alternative fuel to hydrogen is carbon monoxide (CO) that produces carbon dioxide (CO2) and electrons (e2) during oxidation at the anode. The cathode reaction is the same as that in Fig. 13.4b. Due to their high-temperature operation, solid oxide fuel cells can incorporate internal reforming of various hydrocarbon fuels to produce hydrogen and/or carbon monoxide at the anode. Since waste heat is produced at relatively high temperature, solid oxide fuel cells can be used for cogeneration of power and process heat or steam. SOFCs also can be used for distributed (decentralized) power generation and for fuel cell-microturbine hybrids. Such applications are very attractive because they achieve objectives without using highly irreversible combustion. For instance, a solid oxide fuel cell replaces the combustor in the gas turbine shown in the fuel cell-microturbine schematic in Fig. 13.5. The fuel cell produces electric power while its high-temperature exhaust expands through the microturbine, producing shaft power W # net. By producing power electrically and mechanically without combustion, fuel cell-microturbine hybrids have the potential of significantly improving effectiveness of fuel utilization over that achievable with comparable conventional gas turbine technology and with fewer harmful emissions. 13.5 Absolute Entropy and the Third Law of Thermodynamics Thus far, our analyses of reacting systems have been conducted using the conservation of mass and conservation of energy principles. In the present section some of the implications of the second law of thermodynamics for reacting systems are considered.

how this is accomplished for h and u. The case of entropy is handled differently and is taken up in Sec. 13.5. An enthalpy datum for the study of reacting systems can be established by assigning arbitrarily a value of zero to the enthalpy of the stable elements at a state called the standard reference state and defined by Tref 5 298.15 K (258C) and pref 5 1 atm. In English units the temperature at the standard reference state is closely 5378R (778F). Note that only stable elements are assigned a value of zero enthalpy at the standard state. The term stable simply means that the particular element is in a chemically stable form. For example, at the standard state the stable forms of hydrogen, oxygen, and nitrogen are H2, O2, and N2 and not the monatomic H, O, and N. No ambiguities or conflicts result with this choice of datum. ENTHALPY OF FORMATION. Using the datum introduced above, enthalpy values can be assigned to compounds for use in the study of reacting systems. The enthalpy of a compound at the standard state equals its enthalpy of formation, symbolized hf . The enthalpy of formation is the energy released or absorbed when the compound is formed from its elements, the compound and elements all being at Tref and pref. The enthalpy of formation is usually determined by application of procedures from statistical thermodynamics using observed spectroscopic data. The enthalpy of formation also can be found in principle by measuring the heat transfer in a reaction in which the compound is formed from the elements. standard reference state enthalpy of formation consider the simple reactor shown in Fig. 13.1, in which carbon and oxygen each enter at Tref and pref and react completely at steady state to form carbon dioxide at the same temperature and pressure. Carbon dioxide is formed from carbon and oxygen according to C O2 S CO2 (13.6) This reaction is exothermic, so for the carbon dioxide to exit at the same temperature as the entering elements, there would be a heat transfer from the reactor to its surroundings. The rate of heat transfer and the enthalpies of the incoming and exiting streams are related by the energy rate balance 0 Q # cv m # ChC m # O2 hO2 m # CO2 hCO2 where m # and h denote, respectively, mass flow rate and specific enthalpy. In writing this equation, we have assumed no work W # cv and negligible effects of kinetic and potential energy. For enthalpies on a molar basis, the energy rate balance appears as 0 Q # cv n # ChC n # O2 hO2 n # CO2 hCO2 where n # and h denote, respectively, the molar flow rate and specific enthalpy per mole. Solving for the specific enthalpy of carbon dioxide and noting from Eq. 13.6 that all molar flow rates are equal, hCO2 Q # cv n # CO2 n # C n # CO2 hC n # O2 n # CO2 hO2 Q # cv n # CO2 hC hO2 (13.7) Since carbon and oxygen are stable elements at the standard state, hC hO2 0, and Eq. 13.7 becomes hCO2 Q # cv n # CO2 (13.8) Accordingly, the value assigned to the specific enthalpy of carbon dioxide at the standard state, the enthalpy of formation, equals the heat transfer

is obtained when all the water formed by combustion is a vapor. The higher heating value exceeds the lower heating value by the energy that would be released were all water in the products condensed to liquid. Values for the HHV and LHV also depend on whether the fuel is a liquid or a gas. Heating value data for several hydrocarbons are provided in Tables A-25. The calculation of the enthalpy of combustion, and the associated heating value, using table data is illustrated in the next example.

of combustion of an actual combustion process and their relative amounts can be determined only by measurement. Among several devices for measuring the composition of products of combustion are the Orsat analyzer, gas chromatograph, infrared analyzer, and flame ionization detector. Data from these devices can be used to determine the mole fractions of the gaseous products of combustion. The analyses are often reported on a "dry" basis. In a dry product analysis, the mole fractions are given for all gaseous products except the water vapor. In Examples 13.2 and 13.3, we show how analyses of the products of combustion on a dry basis can be used to determine the balanced chemical reaction equations. Since water is formed when hydrocarbon fuels are burned, the mole fraction of water vapor in the gaseous products of combustion can be significant. If the gaseous products of combustion are cooled at constant mixture pressure, the dew point temperature is reached when water vapor begins to condense. Since water deposited on duct work, mufflers, and other metal parts can cause corrosion, knowledge of the dew point temperature is important. Determination of the dew point temperature is illustrated in Example 13.2, which also features a dry product analysis.

per mole of CO2, between the reactor and its surroundings. If the heat transfer could be measured accurately, it would be found to equal 2393,520 kJ per kmol of carbon dioxide formed (2169,300 Btu per lbmol of CO2 formed). b b b b b Tables A-25 and A-25E give values of the enthalpy of formation for several compounds in units of kJ/kmol and Btu/lbmol, respectively. In this text, the superscript 8 is used to denote properties at 1 atm. For the case of the enthalpy of formation, the reference temperature Tref is also intended by this symbol. The values of hf listed in Tables A-25 and A-25E for CO2 correspond to those given in the previous example. The sign associated with the enthalpy of formation values appearing in Tables A-25 corresponds to the sign convention for heat transfer. If there is heat transfer from a reactor in which a compound is formed from its elements (an exothermic reaction as in the previous example), the enthalpy of formation has a negative sign. If a heat transfer to the reactor is required (an endothermic reaction), the enthalpy of formation is positive. Evaluating Enthalpy The specific enthalpy of a compound at a state other than the standard state is found by adding the specific enthalpy change ¢h between the standard state and the state of interest to the enthalpy of formation h1T, p2 hf 3h1T, p2 h1Tref, pref24 hf ¢h (13.9) That is, the enthalpy of a compound is composed of hf, associated with the formation of the compound from its elements, and ¢h, associated with a change of state at constant composition. An arbitrary choice of datum can be used to determine ¢h, since it is a difference at constant composition. Accordingly, ¢h can be evaluated from tabular sources such as the steam tables, the ideal gas tables when appropriate, and so on. Note that as a consequence of the enthalpy datum adopted for the stable elements, the specific enthalpy determined from Eq. 13.9 is often negative. Tables A-25 provide two values of the enthalpy of formation of water: hf 1l2, hf 1g2. The first is for liquid water and the second is for water vapor. Under equilibrium conditions, water exists only as a liquid at 258C (778F) and 1 atm. The vapor value listed is for a hypothetical ideal gas state in which water is a vapor at 258C (778F) and 1 atm. The difference between the two enthalpy of formation values is given closely by the enthalpy of vaporization hfg at Tref. That is, hf1g2 hf1l2 hfg125C2 (13.10) Similar considerations apply to other substances for which liquid and vapor values for ho f are listed in Tables A-25. 13.2.2 Energy Balances for Reacting Systems Several considerations enter when writing energy balances for systems involving combustion. Some of these apply generally, without regard for whether combustion takes place. For example, it is necessary to consider if significant work and heat transfers take place and if the respective values are known or unknown. Also, the effects of kinetic and potential energy must be assessed. Other considerations are related directly to the occurrence of combustion. For example, it is important to know the state of the fuel before combustion occurs. Whether the fuel is a liquid, a gas, or a solid is important. It is also necessary to consider whether the fuel is premixed with the combustion air or the fuel and air enter a reactor separately

the combustion products is assumed to behave as an ideal gas, its contribution to the enthalpy of the products depends solely on the temperature of the products, TP. Accordingly, for each component of the products, Eq. 13.9 takes the form h hf 3h1TP2 h1Tref24 (13.13) In Eq. 13.13, hf is the enthalpy of formation from Table A-25 or A-25E, as appropriate. The second term accounts for the change in enthalpy from the temperature Tref to the temperature TP. For several common gases, this term can be evaluated from tabulated values of enthalpy versus temperature in Tables A-23 and A-23E, as appropriate. Alternatively, the term can be obtained by integration of the ideal gas specific heat cp obtained from Tables A-21 or some other source of data. A similar approach is employed to evaluate the enthalpies of the oxygen and nitrogen in the combustion air. For these h hf 0 3h1TA2 h1Tref24 (13.14) where TA is the temperature of the air entering the reactor. Note that the enthalpy of formation for oxygen and nitrogen is zero by definition and thus drops out of Eq. 13.14 as indicated. The evaluation of the enthalpy of the fuel is also based on Eq. 13.9. If the fuel can be modeled as an ideal gas, the fuel enthalpy is obtained using an expression of the same form as Eq. 13.13, with the temperature of the incoming fuel replacing TP. With the foregoing considerations, Eq. 13.12a takes the form Q # cv n # F W # cv n # F a1hf ¢h2CO2 b 2 1hf ¢h2H2O aa b 4 b 3.761hf 0 ¢h2N2 1hf ¢h2F aa b 4 b 1hf 0 ¢h2O2 aa b 4 b 3.761hf 0 ¢h2N2 (13.15a) The terms set to zero in this expression are the enthalpies of formation of oxygen and nitrogen. Equation 13.15a can be written more concisely as Q # cv n # F W # cv n # F a P ne1hf ¢h2e a R ni1hf ¢h2i (13.15b) where i denotes the incoming fuel and air streams and e the exiting combustion products. Although Eqs. 13.15 have been developed with reference to the reaction of Eq. 13.11, equations having the same general forms would be obtained for other combustion reactions. In Examples 13.4 and 13.5, the energy balance is applied together with tabular property data to analyze control volumes at steady state involving combustion. Example 13.4 involves a reciprocating internal combustion engine while Example 13.5 involves a simple gas turbine power plant.

where AF is the air-fuel ratio on a molar basis and AF is the ratio on a mass basis. For the combustion calculations of this book the molecular weight of air is taken as 28.97. Tables A-1 provide the molecular weights of several important hydrocarbons. Since AF is a ratio, it has the same value whether the quantities of air and fuel are expressed in SI units or English units. Theoretical Air The minimum amount of air that supplies sufficient oxygen for the complete combustion of all the carbon, hydrogen, and sulfur present in the fuel is called the theoretical amount of air. For complete combustion with the theoretical amount of air, the products consist of carbon dioxide, water, sulfur dioxide, the nitrogen accompanying the oxygen in the air, and any nitrogen contained in the fuel. No free oxygen appears in the products.

Thermodynamics 13

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