Recalculation of Air Turborocket Performance

第一工業大学研究報告 41 第25号（2013）pp.41－47 Recalculation of air turborocket performance Recalculation of air turborocket performance Recalculation of air turborocket performance Koichi SUZUKI＊ Koichi Suzuki＊ Koichi Suzuki＊ The turborocket engine has been proposed for use in very-high-speed vehicles, such as Earth-to-orbit The turborocket launchers. engine has The been pure proposed ramjet forengine use incan’t very-high-speed work at low vehicles,Mach numbers, such as whileEarth-to-orbit the turbojet launchers. engine hasThe poorpure performance ramjet engine at highcan’t Machwork numbersat low Mach because numbers, of the limitationwhile the turbojetof turbine engine inlet temperature.has poor performance The turborocket at high engineMach numbers improves because weak pointsof the oflimitation both engines, of turbine it has inlet one temperature. or two stage The fans turborocket which work engine at low improves Mach numbers weak points and turbinesof both engines, which use it hasindependent one or two gas stage from fansfree streamwhich workair. The at turborocketlow Mach numbers engine andhas beenturbines introduced which use in theindependent standard gastextbook from freeof Dr. stream Jack L.air. Kerrebrock’s The turborocket “Aircraft engine Engines has andbeen Gasintroduced Turbines” in the , standardbut unfortunately textbook of thereDr. Jack are L. someKerrebrock’s misunderstandings “Aircraft Engines and miscalculationand Gas Turbines” of the ,engine but unfortunatelyperformance. Thethere engine are someperformance misunderstandings of his book is andtoo high.miscalculation The text bookof the is enginenot revised performance. though it Theused engine worldwide, performance so this paperof his is book one ofis thetoo proposalhigh. The of text correction book is ofnot chapter revised 10.8 though of the it book. used worldwide, so this paper is one of the proposal of correction of chapter 10.8 of the book. １．The Air Turborocket There１．The are Air two Turborocket type of the air turborocket , one version uses two liquid propellants, a fuel andThere an are oxidizer, two type which of the are air pumped turborocket to high , one pressure version and uses burned two liquid in a propellants,gas generator a fuel, as indicatedand an oxidizer, in the upperwhich halfare pumpedof figure to10.26 high ( thispressure figure and are burned adopted in from a gas his generator book ) . The, as fuel-richindicated combustion in the upper gas half is expandedof figure 10.26 through ( this a turbine,figure are which adopted drives from the his air book compressor, ) . The andfuel-rich then combustioncombusted withgas is the expanded compressed through air. a turbine, which drives the air compressor, Inand another then combusted version, shown with the in compressedthe lower half air. of figure 10.26, liquid hydrogen is pumped toIn highanother pressure, version, vaporized shown inand the heated lower by half heat of exchanger,figure 10.26, and liquid then hydrogen expanded is through pumped a turbine,to high pressure, which drives vaporized the air and compressor, heated by andheat then exchanger, combusted and withthen theexpanded compressed through air. a turbine, which drives the air compressor, and then combusted with the compressed air. *Visiting professor, Daiichi Institute of Technology, Ueno-campus, Tokyo *Visiting professor, Daiichi Institute of Technology, Ueno-campus, Tokyo This document is provided by JAXA. 42 第一工業大学研究報告第25号（2013） The analysis of either version of the air turborocket proceeds similarly to that of the turbojet (refer to previous section of the book), but some qualitative differences arise from the separation of the compressor and turbine flows. Thus、 the nozzle pressure and temperature ratios are given by １２１１Ｍ２７ �⁄ �� ｄｃ (1)(10.34) �� Where�� p � ; �pressure,� M ; Mach� � � number,� � �� ; specific heat ratio, = , = diffuser pressure loss, γcompressor pressure ratio,�� =��⁄�� afterburner� ��⁄�� pressure loss. Suffix t �means� � �� total⁄�� condition, 7 means point number.�� ⁄The�� equation number (10.34) is original equation number of his book. And = (2)(10.35) �� Where ��= � � is� afterburner � � exit temperature ratio. So that �for� �an�� ⁄ideally�� expanded nozzle (3)(10.36) � � ��⁄� �� And = �� ⁄� �� ⁄�� Where is inlet temperature ratio.The last equality applies only for the ideal cycle, without�� diffuser⁄�� or combustor pressure losses. The thrust per unit of airflow is given by (4)(10.37) � �� The �specific� � � � impulse�� is� �� (5)(10.38) � �⁄ �� ⁄� �� ⁄�� Where F �; thrust, I ; specific impulse, ; air mass flow rate, ; propellant mass flow rate, ; free stream sound velocity.�� To compute�� the thrust and the specific impulse we must then determine the afterburner exit temperature ratio , the compressor temperature rise , and the ratio of propellant to air mass flows.� The� interrelationship of these is different�� for the bipropellant and expander versions, but they have some features in common. �� In most cases, the performance of the turborocket will be limited by the pressure This document is provided by JAXA. SUZUKI：Recalculation of air turborocket performance 43 ratio available from the compressor, which is set by the available turbine power, so it will be desirable to operate the turbine at as high an inlet pressure as is practical. The limit will be set by stresses and temperatures in the turbine and in the combustion chamber. Thus, we regard and as design parameters of the turborockets, in the same sense that is a design�� parameter�� for the turbine engines. The available compressor�� temperature rise is then determined by the turbine –compressor work balance: , (6) �� Where , being specific heat ratio of compressor and turbine gases. The first term on the left�� is�� the power required to pump the propellants, being a mean density for the propellant mixture. Normally this term is small in relation�� to the compressor work, but it will be retained for now. Solving for the available compressor temperature rise, we get �� 1� � �� 1� � �� (7) �� Introducing � � the� � perfect 1 � � gas� law;� � � � � �, 1� , �� 1�⁄� m� � ρ� γ � 1 p�� c��T��τ� � 1� � c��T��θ� � θ�� c��T� � � 1� �� m� � ρ� γ p� ��⁄� �� 1 1 �� 1 � � � �� 1�� (9) �� ⁄� � �� which can be simplified considerably by noting that the second term in the denominator will normally be much less than unity, and that is much greater than unity. Thus, ��⁄�� (10)(10.39) �� Above� �analysis 1 � �is induced� by Dr.� J.K.Kerrebrock, but unfortunately we can’t use the perfect gas law, because in equation (6) is the inlet pressure of propellant, not of air. Let is the inlet pressure�� of propellant, equation (6) becomes � � (6)’ �� This document is provided by JAXA. 44 第一工業大学研究報告第25号（2013） (7)’ �� 1� � �� 1� �� (9)’ �� ⁄ � � � � Which can be�� simplified�� considerably by noting that the second term in the denominator will normally be much less than unity, and that is much greater than unity. Thus, ��⁄�� (10)’ �� Equation � � 1 �(10) �and �(10)’ seems �to � 1be � the� same,� but it’s� occasional case caused by neglecting unity. Equation (9)’ is correct and equation (9) is false. 2. H2-Expander ATR There is a release of chemical energy only in the afterburner in H2-expander ATR, only an overall heat balance is required, in the form � �� m� c �T � T � � m� h (11)(10.43) �� T� �� Where� θ �is θheat release of hydrogen. This relation may be regarded as determining if is prescribed. The afterburner temperature sets the hydrogen fuel flow, h�� m�and�⁄ m�then� theθ� temperature and pressure in the compressor drive turbine determine the compressor pressure ratio. Thus, the thrust per unit of air flow can be determine and also specific impulse. We determine some values of parameters as follows. , , , (assumed height of 30000m),c�� 1.00� kJ⁄ kg� c ��(assumed� 1�.� kJ height⁄ kg� of� �30000m), 1.� T � � ��.��=141947.2 ��, � 301.� m, ⁄ � h�� kJ⁄ kg Theθ �calculated� 10.0 θ values� � �.0 arep�� described� 100a�m �in10.13MPa the Table� -1, and plotted in figure 1 and 2. Table 1 Performance of Air Turborocket M 1 2 3 4 5 6 1.2 1.8 2.8 4.2 6 8.2 0.014 0.0131 0.0115 0.00929 0.0064 0.00288 � � 1.817 1.51 1.287 1.155 1.074 1.024 �� ⁄�� 8.453 6.488 4.73 3.276 1.98 0.7959 � F'/(maa0)� 4.202 3.621 3.01 2.3 1.499 0.6366 �⁄ �� 18575.6 15237 12653.9 10848.9 9518 8502.1 ��⁄ �� Isp' 9233.9 8503.9 8052.4 7616.8 7205.8 6800.4 �� This document is provided by JAXA. SUZUKI：Recalculation of air turborocket performance 45 9 8 7 6 5 � 4 �� 3 F'/(maa0)�⁄ �� 2 ��⁄ �� 1 0 0 1 2 3 4 5 6 7 M0 Figure 1 ThrustFigure per unit 1 Thrust of airflow per unit for of air airflow turborocket for

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