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Appendix A: Efficiency at maximum power An interesting characteristic of most endo-reversible engines is the maximum power efficiency; i.e., rﬃﬃﬃﬃﬃﬃﬃ TL (A.1) η¼1 TH For many years after Eq. (A.1) appeared in Curzon and Ahlborn (1975)’s paper, the efficiency in Eq. (A.1) carried the subscripts “CA” denoting the initials of the authors last names. Later, it turned out that Eq. (A.1) had been presented about two decades earlier by Chambadal and Novikov. There is however evidence [1] that the expression for the efficiency at maximum power had already been derived by James Henry Cotterill, Professor of Applied Mechanics in the Royal Naval College, in the late 19th century. The maximum power efficiency can be found in the second edition of the Cotterill’s textbook [1], Chapter IV, pages 100–102. The author was unsuccessful in locating the first edition of the book published in 1877 to confirm whether Eq. (A.1) was first given in the earliest edition. Nevertheless, important to remember is that it would be inappropriate to refer Eq. (A.1) as Chambadal-Novikov-Curzon-Ahlborn efficiency.

Appendix B: Effect of fuel type on SEG To examine the effect of the fuel type on SEG, the efficiency and the specific entropy generation are calculated and compared for the gas turbine cycle studied in Chapter 8 operating on hydrogen, propane, methanol, and ethanol. At a given pressure ratio and TIT, the specific entropy generation is quantitatively different depending on the type of the fuel burnt. The highest and lowest values of SEG (measured in J/mol K) are obtained for propane (C3H8) and hydrogen, respectively. A fuel with a greater heating value would yield a higher SEG. The heating values of the fuels and the minimum specific entropy generation of the cycle at TIT ¼ 1100 K are compared in Table B.1. Note that if the calculations are performed on mass basis the minimum SEG would still correlate with the heating value. In this case, 189

190

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Table B.1 Comparison of the heating value and the minimum specific entropy generation of a gas turbine cycle (TIT ¼ 1100 K) for five different fuels. Minimum SEG Fuel HV (kJ/mol) (J/mol K)

H2 CH3OH CH4 C2H5OH C3H8

241.8 675.9 802.3 1277.5 2043.9

536 1676 1956 3187 5075

hydrogen with the highest heating value (measured in kJ/g) among the five fuels would yield the greatest specific entropy generation, whereas methanol with the least heating value would lead to the lowest SEG.

Appendix C: Determination of ξ at minimum Gfm An equation can be derived for the reaction advancement that minimizes the function Gfm but maximizes the entropy generation, see Eq. (10.10), for a mixture of k ideal gases, i.e., Eq. (10.19). Substituting the relation f

sj ðT , pÞ ¼ s0j + sT , j R lny j

(C.1)

for the entropy of species j in Eq. (10.19) yields Gmf ðξÞ ¼

k X

h i f nij + aj ξ hj ðT Þ Ts s0j + sT , j R ln y j R lnp (C.2)

j¼1

where s0 denotes the specific entropy at the standard temperature and pressure, and sT is the entropy change due to the difference between the temperature T and the standard temperature. The model fraction of species j is defined as f

yj ¼

f n j nij + aj ξ ¼ n f ni + aξ

(C.3)

where n fj and n f are substituted from Eqs. (10.15) and (10.17), respectively. Substituting Eq. (C.3) into Eq. (C.2), Gfm can be described as a function of ξ only.

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Appendices

Gmf ðξÞ ¼

k X j¼1

" ! !# i n + a ξ j j R lnp nij + aj ξ hj ðT Þ Ts s0j + sT , j R ln i n + aξ (C.4) Gfm,

dGfm/dξ ¼ 0

At the minimum one must solve to get !a j ( ) k h i X nij + aj ξ 0 aj aj hj ðT Þ Ts sj + sT , j + Ts R ln i + Ts R lnp ¼ 0 n + aξ j¼1 (C.5) Upon simplification and rearrangement, we obtain !aj ( ) k k n h io Y nij + aj ξ 1 X ¼ exp aj hj ðT Þ Ts s0j + sT , j a lnp i + aξ n T R s j¼1 j¼1 (C.6) For a reactive system with a known initial state and composition that interacts with its surrounding that is at temperature Ts, the only unknown in Eq. (C.6) is ξ. If ξ at minimum Gfm happens to be close to ξeq (as was seen in Fig. 10.6) the composition of the mixture at the final state may then be readily obtained from Eq. (C.3) and using ξ determined from Eq. (C.6).

Reference [1] J.H. Cotterill, The Steam Engine Considered as a Thermodynamic Machine, A Treatise on the Thermodynamic Efficiency of Steam Engines, second ed., E & F. N. Spon, London, 1890.