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# Thermodynamic Cycle Analysis for Propagating Detonations

Thermodynamic Cycle Analysis for Propagating Detonations

## this detailed energy

this detailed energy balance and the thermodynamic system approach demonstrates theself-consistency of our formulation of the FJ detonation cycle.Thermal efficiencyThe FJ cycle can be used to define an upper bound on the efficiency of devices using apropagating detonation as the combustion step. The thermal efficiency for the conversion ofchemical energy into mechanical work is the ratio of the net work done to the specific heatof combustion of the mixture. From the previous discussion, for FJ cycle this isη th = w netq c= h 1 − h 5q c. (1)The specific heat of combustion q c is computed as the enthalpy difference between the reactantsand the products at initial pressure and temperature: q c = h 1 − h 6 .We first investigate the values of the thermal efficiency for a perfect gas model by representingthe detonation process using the one-γ model of detonation, 14 which relates the CJMach number to the specific heat of combustion.M CJ = √ H + 1 + √ H where H = (γ2 − 1)q c2γRT 2(2)[η th = 1 − C pT 1q c1M 2 CJ( ) γ+1 ]1 + γM2 γ CJ− 11 + γThe thermal efficiency is represented in Fig. 3 as a function of the CJ Mach number fortwo values of the specific heat ratio γ representative of the products of fuel-oxygen and fuelairdetonations. The thermal efficiency increases with increasing CJ Mach number, whichincreases 14 with the specific heat of combustion q c . A higher heat of combustion increases thepressure ratio through the detonation wave significantly more than the temperature ratio.This means that the detonation products undergo a stronger expansion between states 4 and5, which reduces the temperature at state 5. Thus, less heat is rejected during process 5–1and the efficiency is higher. Figure 3 also shows that the variation of the thermal efficiencydepends strongly on the value chosen for γ. At constant CJ Mach number, a lower valueof γ in the detonation products yields a lower efficiency. The parameter γ − 1 controls theslope of the isentrope 4–5 in the pressure-temperature plane. Although the pressure ratioP 4 /P 5 varies with γ, the main effect of decreasing γ is to increase the temperature at state5 so that more heat is rejected during process 5–1. The result of Eq. 3 is identical to theresult obtained by Heiser and Pratt, 3 who computed the thermal efficiency by calculatingthe entropy increments associated with each process in the detonation cycle. However, the(3)4 of 14

al. 8 In reality, one- or two-γ models of these cycles cannot correctly capture all the features ofnumerical values we obtain are lower than those given in Heiser and Pratt 3 because theyused a value of γ = 1.4 corresponding to the reactants, whereas we use values of γ equal to1.1 or 1.2 more representative of the detonation products. A more realistic cycle analysis fora perfect gas involves using the two-γ model of detonations, 14 which was applied by Wu etdissociation-recombination equilibria and temperature-dependent properties. The thermalefficiency was calculated based on Eq. 1 using realistic thermochemistry 15 for hydrogen,ethylene, propane, and JP10 fuels with oxygen and air, assuming all chemical states involvingcombustion products are in equilibrium. The results are significantly influenced by thevariation of the specific heat capacity with temperature in the detonation products and thedissociation and recombination processes.The influence of equivalence ratio on the thermal efficiency is shown in Fig. 4. Thethermal efficiency for fuel-air mixtures is maximum at stoichiometry, whereas it is minimumfor fuel-oxygen mixtures. Fuel-air mixtures generate much lower CJ temperatures than fueloxygenmixtures. Because of their low extent of dissociation, fuel-air mixtures follow thesame trends as the perfect gas and yield a maximum efficiency when the energy release ismaximized near stoichiometry. On the other hand, fuel-oxygen mixtures are characterizedby high CJ temperatures (in particular near stoichiometry), which promote endothermicdissociation reactions. Although the radicals created by the dissociation reactions startrecombining during the subsequent expansion process, the temperature in the detonationproducts of fuel-oxygen mixtures remains high and only partial recombination occurs. Theenergy stored in the partially dissociated products at state 5 is proportionally higher forfuel-oxygen than for fuel-air mixtures, which reduces the thermal efficiency of fuel-oxygenmixtures relative to fuel-air mixtures. This effect increases with increasing CJ temperatureand was confirmed by investigations of the influence of nitrogen dilution, 13 which showedthat the thermal efficiency increases with increasing nitrogen dilution. The thermal efficiencyis also found to decrease with decreasing pressure 13 due to the increasing importance ofdissociation at low pressures, and with increasing initial temperature 13 due to the associateddecrease in initial mixture density and CJ Mach number.Although stoichiometric fuel-oxygen mixtures have a higher specific heat of combustionthan fuel-air mixtures, Fig. 4 shows that fuel-air mixtures have a higher thermal efficiency.This is attributed mainly to dissociation phenomena, but also to the higher value of theeffective ratio of specific heats γ in the detonation products of fuel-air mixtures. 13 Note thatfuel-oxygen mixtures still generate 2 to 4 times as much work per unit mass of mixture asfuel-air mixtures because of their larger specific heat of combustion.5 of 14

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