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For steady horizontal motion of a body with one inlet and one exit (like a jet engine) the momentum balance, in the direction of motion, reduces to (see Eq. 5.20 and 17.12): Tm(v v)(p p)A (p p)A (17.15) eie0ei0i where the approximation to neglect the additional mass flow rate of fuel is introduced. The propulsion efficiency, defined as for rockets, becomes: viv 2 v (17.16) Tv Tv m(v v)2 pep0 m(v v)v e1i p 1 2e ei2e v m(v v)v m(v v)2 where the approximation to consider the inlet speed equal to the flight speed has been introduced. Similarly to chemical rockets, air-breathing engines (turbojets) are heat engines that transform the heating power of a fuel into mechanical energy (the kinetic energy of the exiting jet minus the one at the inlet), and thus an energy efficiency may be defined by: 1mv2 1mv2 e 2 e 2 i (17.17) mf hP where hP is the heating power of the fuel/oxidiser mixture per unit mass of fuel. Notice that the energy efficiency is based on thermal-to-mechanical conversion, and the propulsion efficiency is based on mechanical-to-propulsive ratio, thus, the efficiency using the fuel for propulsion is ep. The overall energy balance of the air-breathing engine may help to identify every term; in axes fixed to the surroundings: 1 e , may be obtained with a large v and small flow rate or vice versa, but the , what Notice that the thrust, mv propulsion efficiency, p=2/(2+v/v), shows that it is better to have a small v and a large m has been implemented in practice, where turbofans of high bypass have superseded simple turbojets. with: Hchem Wprop Kjet Hjet (17.18) H mh, W Tv, K 1m(vv)2, H mc(TT) (17.19) chem f p prop jet 2 e jet p e i being the chemical enthalpy released, Wprop H chem exit jet, and H jet the propulsive power, K jet the thermal enthalpy of the exit jet. the kinetic energy of the It is important to compare air-breathing propulsion with rocket propulsion for the same fuel isoftheorderof2%of m (17.20) expenditure (notice that for rockets mf m The quotient of respective thrusts is: ). , whereas for turbojets mf Tturbojet pep0 mair(ve,turbojet vi,turbojet)viv ve,rocket T m v v v rocket fuel ,rocket e,rocket e,turbojetPDF Image | THERMAL MACHINES AND HEAT ENGINES
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