10/10/02 ME3324 Collaborative Problem 5 1Given: A ramjet engine operating at steady state with known states at the inlet (ambientconditions, air approach velocity, and inlet area), combustor (air-to-fuel ratio, fuel LHV,and combustor efficiency), and exit (exit area and exit temperature).DiffuserCombustorNozzle1234ADAnVaVjmfFind: The energy equations for each engine component and an overall efficiency metricfor the engine.Solution:Diffuser Analysis:12VaADUse system defined by red outline in sketch. Assuming steady flow, the mass balance forthe diffuser turns out to be:eimm(1)The full energy balance is:e2ei2igz2Vhmgz2VhmWQdtdE00 0(2)Simplifications have been made by assuming steady flow, the diffuser is adiabatic, andthe potential energy change is negligible, as well as recognizing that there is no work andthere is only one stream. Further algebraic manipulation leads to:)T(f2Vhh2Vhm2Vhm2211222222111(3)In coming to the final boxed solution, eqn. (1) has been used to cancel out the mass flowrates, and the velocity at state 2 has been ignored. This is based on the assumption that10/10/02 ME3324 Collaborative Problem 5 2the diffuser is well made, meaning that all of the kinetic energy has been converted intopressure and temperature rise (enthalpy). Assuming air to be an ideal gas, thetemperature at state 2 can be fixed with the knowledge of h2 alone and Table A-17 in thebook. An alternate approach would be to use specific heats to find h1 and h2.Combustor Analysis:23Q2m3mThe combustor can be treated as a constant cross-section area device that allows heattransfer to the flow stream. Writing an energy balance for the combustor gives:e2ei2igz2Vhmgz2VhmWQdtdE00(4)This equation can be reduced to:airin23mQhh(5)Since h2 is just a function of the combustor inlet temperature (found from diffuseranalysis), and the ratio of airinmQ can be found from the analysis of the fuel combustion,h3 (and hence the exit temperature) can be solved for. Unfortunately, not all the energyin the fuel is used to raise the temperature of the air. This is characterized through thecombustor efficiency, c, which is defined as:LHVmQfuelin energy air fer toheat transfuelincomb(6)The mass-flow-rate of the fuel is then found from the air-to-fuel ratio, AF, and the airmass flow rate that is found using the diffuser inlet conditions.10/10/02 ME3324 Collaborative Problem 5 3AFAVAFmm1daairfuel(7)The inlet air properties can be found by assuming standard temperature and pressure atthe inlet of the diffuser. The final energy equation combining eqn’s (5) – (7) is:AFLHVhmAFmLHVhmLHVmhhcomb2airaircomb2airfuelcomb23(8)Nozzle Analysis:43VjANThe nozzle analysis is very similar to the diffuser analysis in terms of assumptions andthe energy balance. The main difference is that the inlet and exit temperatures are known(T3 from the combustor analysis and T4 is given, so h3 and h4 are known) and the jetexhaust velocity needs to be found. Proceeding in this way, the mass balance yields:43mm(9)The energy balance is:e2ei2igz2Vhmgz2VhmWQdtdE00 0(10)The assumptions are steady flow, adiabatic, no work, and no change in potential energy.Further algebra manipulation gives:432424442333hh2V2Vhm2Vhm (11)This time the nozzle inlet velocity is assumed to be negligible, and the quantity of interestto solve for is the jet exhaust velocity. Finally:10/10/02 ME3324 Collaborative Problem 5 4 434hh2V (12)where:j4VV (13)Overall Efficiency:There are multiple ways to define efficiency for a ramjet engine, depending on whataspect of the engine you are interested in. In terms of overall performance, it iscustomary to write the efficiency in terms of the thrust power developed (what you wantfor propulsion) divided by the LHV of the fuel (what it costs you). This is: LHVmVVVmLHV fuelpowerthrust fuelaajairo(14)Note that the thrust power shown in the numerator is derived by considering a forcebalance on the engine, which is beyond the scope of this class. For more information, see“Propulsion Systems” by A. N. Hosny, University of South Carolina Press, Columbia,SC, 1974, or any other propulsion systems