Appendix A Performance of the Ericsson Cycle Chapter 3 shows that intercoolers and reheat combustors can improve the per­ formance of gas turbines. Cooling the air during the compression process with intercoolers reduces the power requirement of the compressor stages. Heat­ ing the gas during the expansion process with reheat combustors increases the power output of the turbine stages. Thus, intercoolers and reheat combustors can improve thermal efficiency and cycle specific power. A gas-turbine cycle with an infinite number of intercoolers and an infinite number of reheat com­ bustors is known as an Ericsson cycle, after the Swedish inventor John Ericsson. This appendix shows that the thermal efficiency of an Ericsson cycle with ideal components approaches the Carnot efficiency, which is the maximum thermal efficiency achievable by any heat engine. To find the thermal efficiency of an Ericsson cycle, we will derive expressions for the heat inputs of the combustors, the power inputs to the compressors, and the power outputs of the turbines. Then, we will let the number of compressor stages, turbine stages, and combustors approach infinity and show that the thermal efficiency approaches the Carnot efficiency. Thermal efficiency can be calculated as the ratio of the cycle specific power to the cycle specific heat-input rate: W' (A.I) 'fJTH = Q1 ' where 'fJT H Thermal efficiency (~); Q' Cycle specific heat-input rate (~); and W' Cycle specific power (~). Cycle specific power can be calculated using Equations 3.5 and 3.6. Cycle specific heat-input rate is the sum of the specific heat-input rates of the n 235 236 A Performance of the Ericsson Cycle combustors in the cycle: (A.2) j The specific heat-input rate of each combustor is the heat-input rate to the combustor divided by (roughly 1 ) the enthalpy flow rate into the gas turbine: Q' QHj (A.3) Hj = (mcpT)o ' where Specific heat-input rate of Combustor j (-); Heat-input rate to Combustor j (W); Roughly the enthalpy flow rate into the gas tur­ bine (W); Mass flow rate (kgj s); Cp = Specific heat capacity evaluated at a constant pressure (J jkg-K); and T Total temperature (K). With an ideal regenerator (100% effective), the heat-input rate to each of the combustors (including the first combustor) is equal to the power output of each successive turbine stage. Thus, thermal efficiency is ~7=1 Wej (AA) TJTH = 1 + ",n W' 6j=1 Ej where ~7=1 Wej Sum of the specific powers of the n compressor stages (-); and ~7=1 W.b Sum of the specific powers of the n turbine stages (-). In this approximate analysis, we assume a constant specific heat capacity of the working fluid (air and combustion products for open-cycle gas turbines). The specific heat capacity of air actually increases by about 10% for every 600 K of temperature increase. For a constant specific heat capacity and ideal compressor stages (100% efficient), the sum of the specific powers of the n compression stages is ~ [(R/CP)] ~ Wej = n 1 - r-n- , (A.5) j=l where r Cycle pressure ratio, the ratio of the outlet pres­ sure of the last compressor stage to the inlet pres­ sure to the gas turbine (-); and R G as constant for air (J jkg- K) . 1 As explained in Chapter 3, the denominator in Equation A.2 is only roughly the enthalpy flow rate into the gas turbine because specific heat capacity varies with temperature. A Performance of the Ericsson Cycle 237 Similarly, the sum of the specific powers of the n ideal (100% efficient) turbine stages is """,n , [ -(R/CP)] L..tWEj=nT 1-r n , (A.6) j=l where T' = Cycle temperature ratio, the ratio of the outlet temperature of each of the n combustors to the inlet temperature to the gas turbine (-). We insert the expressions for the specific powers of the compressor and tur­ bine stages (Equations A.5 and A.6) into our expression for the thermal effi­ ciency (Equation A.4) and take the limit as the number of compressor stages, turbine stages, and combustors approaches infinity: [l_r(R/,;P)] 1 'TJTH = lim 1 + --''"""[-----,('"''R,-;-/C....:..,..)"] = 1 - T' . (A.7) n-+oo T' 1 - r~ This is the Carnot efficiency (see Equation 3.3). References [1] Babcock & Wilcox Co., "Steam, Its Generation and Use," New York, 1963. [2] Bahnke, G. D. and C. P. Howard, "The Effect of Longitudinal Heat Con­ duction on Periodic-Flow Heat Exchanger Performance," Transactions of the ASME, Journal of Engineering for Power, Vol. 86, 1964. [3] Bannister, R. L., N. S. Cheruvu, D. A. Little, and G. McQuiggan, "De­ velopment Requirements for an Advanced Gas Turbine System," AS ME Paper 94-GT-388, AS ME New York, New York, 1994. [4] Beck, D. S. 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