## Maximum Thermal Efficiencies

The thermal efficiency of a heat engine is determined by the amount of work the engine can perform with the thermal energy supplied to the system. The heat, Qin, is released from the fuel when it is oxidized and is transferred to the working fluid (in the case of an external combustion engine). The expansion of the working fluid (the combustion gases themselves in an internal combustion engine) is harnessed by machinery and converted to mechanical work, Wnet. Equation (3.34) is the general expression for the thermal efficiency, nth, of heat engines, which was presented earlier in Eq. (3.20).

Q in

The maximum thermal efficiency that can be achieved by a heat engine is given by the theoretical Carnot cycle, which is thermodynamically reversible. Equation (3.24), from Section 3.1.4, shows that nth,Carnot, the thermal efficiency of reversible heat engines, depends on the ratio of the low (TL) and high (TH) temperatures in the thermodynamic cycle. Because the low temperature is usually fixed at the ambient condition, the efficiency is therefore determined by the highest temperature in the cycle: the higher the temperature, the higher the efficiency.

To reach the highest possible temperature, however, the fuel loses a portion of its chemical energy to irreversible processes that occur during combustion. For example, in the adiabatic combustion of methane burning with excess air, 35% of the available work potential of the fuel is consumed in reaching the maximum temperature even before the thermal energy can be converted to do work (Qengel and Boles, 1998).

Electrochemical cells such as storage batteries and fuel cells, on the other hand, operate at constant temperatures with the products of the reaction leaving at the same temperature as the reactants. Because of this isothermal reaction, more of the chemical energy of the reactants is converted to electrical energy instead of being consumed to raise the temperature of the products; the electrochemical conversion process is therefore less irreversible than the combustion reaction. In the electrochemical cell, none of the criteria that define heat engines (Section 3.1.4) is satisfied, so the Carnot cycle efficiency, which limits the maximum work to the highest temperature of the cycle, is irrelevant to electrochemical cells. Instead, the maximum work for an electrochemical cell, Wmax,cell, is equal to the change in the Gibbs function (or Gibbs energy), AG, between products and reactants.

(The derivation is presented later in Section 3.4.4). The work, which is done by the movement of electrons through a difference in electrical potential, is denoted Wcell in Eq. (3.36). In electrical terms, the work done by electrons with the charge neF moving through a potential difference, E, is:

In Eq. (3.36), neis the number of electrons transferred per mole of fuel and F is the charge carried by a mole of electrons, which is Faraday's number (96,485 C/mole-1). Figure 3.9 illustrates how an electron does work as it moves through a potential difference.

To make a direct comparison between heat engines and electrochemical cells, the First-Law-based efficiency is used, with Eq. (3.36) substituted into Eq. (3.34) for Wnet. The higher heating value of the fuel (HHV; water in the combustion products is in the condensed form) replaces Qin.

Wcell neFE

The maximum thermal efficiency of an electrochemical cell is given at the open-circuit voltage, E°, the equilibrium condition in which no current is being drawn from the cell.

The value of E° can be determined by relating Eq. (3.35) to Eq. (3.36) and by using the tabulated Gibbs energy data in thermodynamic texts. For example, E° = +1.23 V for a hydrogen-oxygen fuel cell, so its maximum thermal efficiency (at 25°C, 1 atm) is nth = 2 x 96,485 x 1.23/285,840 = 0.83. (The inefficiency is attributed to the entropy generated from the chemical reactions.) For a Carnot cycle heat engine to match this thermal efficiency, the high temperature of the cycle would have to be 1480°C (with the low temperature being 25°C).

In Fig. 3.10, the reversible work for an electrochemical cell is compared to that of a reversible heat engine. The electrochemical cell in this example is a fuel cell that uses hydrogen, H2, and oxygen, O2, to produce water vapor, H2O. As the temperature increases, the change in the Gibbs energy of the reaction decreases, so from Eq. (3.35), the maximum work output from the fuel cell also decreases. In this case, the Gibbs energy of the formation of water vapor is -228.582 kJ/mol at standard temperature and pressure (298.15 K and 1 atm) and decreases to -164.429 kJ/mol at 1500 K. The reversible work of the heat engine, using the HHV of H2 as the source of heat, increases with temperature because the Carnot cycle efficiency increases. Below 950 K, the hydrogen fuel cell converts more of the chemical energy of its reactants (H2 and O2) to work, but above 950 K, the Carnot engine produces more work from the combustion of H2.

Thermal efficiency of automobile engines is usually calculated in terms of power, so the heat input is written as a rate according to the flow rate of fuel. By incorporating the flow rate, the thermal efficiency includes a factor related to the completeness of the combustion of fuel. For fuel cells, the analogous concept to completeness of combustion is fuel utilization, a measure of the fuel consumed to produce an electrical current. In electrical terms, it is called current efficiency, nI, and is given in Eq. (3.39). Its inverse is the fuel stoichiometry, which is the amount of fuel fed to the cell compared to amount the cell requires to provide the electrons demanded.

nI -neFNfuel fu el stoichiometry (3.39)

S 50 IT

S 50 IT

0 200 400 600 800 1000 1200 1400 1600 Temperature [K]

FIGURE 3.10 The reversible work produced by a H2/O2 fuel cell is greater than that of a Carnot engine at temperatures below 950 K. At higher temperatures, the Carnot engine is able to convert more of the HHV of H2 (285.840 kJ/mol) into work. The data for the standard Gibbs energy of formation for water vapor was taken from Lide (1995, pp. 5-64).

0 200 400 600 800 1000 1200 1400 1600 Temperature [K]

FIGURE 3.10 The reversible work produced by a H2/O2 fuel cell is greater than that of a Carnot engine at temperatures below 950 K. At higher temperatures, the Carnot engine is able to convert more of the HHV of H2 (285.840 kJ/mol) into work. The data for the standard Gibbs energy of formation for water vapor was taken from Lide (1995, pp. 5-64).

In Eq. (3.39), I is the current in A and Nfuel is the flow rate of fuel in mol/sec. Assume that hydrogen is used as fuel and that the current efficiency is 83%. One mole of hydrogen contains two moles of electrons (ne = 2), and a current efficiency of 83% means that 83% of the hydrogen is converted to electricity. The remaining 17% either leaves the cell without reacting or reacts non-electrochemically (without contributing its electrons to the cell current). For fuel cells with inlet and outlet flows, the fuel stoichiometry is greater than one (utilization less than 100%), to provide excess fuel to the sections of electrode at the end of the flow channel, maintaining a more uniform distribution of performance over the electrode. The excess fuel that exits the cell may be recycled into the fuel cell (if nonreactive components are absent) or may be chemically reacted to produce heat. Fuel cells without an outlet (with a dead end) may have utilizations of 100% because the fuel that is fed to the cell is completely consumed.

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