Lecture 25: Basic Turbomachine Performance

16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 25: Basic Turbomachine Performance Turbopump Pressurization Systems 1. Cycles For higher performance, mechanical pumps must be used to feed the combustion chamber. In turn, these pumps require drive power, which is always provided by turbines using excess thermal energy in the propellants (although electrical motors have been considered for small rockets). The manner in which hot gas is provided to drive the turbines serves to distinguish among several pressurization cycles, of which the most important are summarized in Fig. 1. (From Ref. 41). The most common of these are the bi-propellant gas generator (G.G.) cycle, the expander cycle and the staged combustion cycle. (a) The Gas Generator Cycle The GG cycle was used in the F1 engine, and is also in use in the Delta II, Atlas and Titan rockets. In this cycle, a small fraction of the pressurized oxidizer and fuel are diverted to a medium-temperature burner (Gas Generator), which produces typically very fuel-rich gas to drive the turbine or turbines. These are designed with a large pressure ratio, and their exhaust is either dumped overboard, or injected at some point into the main nozzle to provide some extra thrust. Nevertheless, this cycle is inherently somewhat lossy, in that the turbine gas is not fully utilized in the main combustor. On the other hand, the power control is relatively straightforward, and there is little interaction of the feed system with the rest of the rocket. Any propellant combination can be used, all power levels are suitable, and any desired pressure level can be obtained although the Isp loss increase with pressure (1.5-4 sec per 100 atm). The mechanical power required to drive the pumps is • PP = mox ∆POP ηoP ρox • + mF ∆PFP ηFP ρ F (1) where ∆POP and ∆PFP are the pressure rise in the oxidizer pump (OP) and fuel pump (FP), respectively which have efficiencies ηOP , ηFP . Also ρox , ρF are the liquid densities. • If the gas generator mass flow rate is mGG , the (single) turbine power is • ' PT = mGG ηT cP' TnηTT (2) in which ηT is the turbine isentropic efficiency (60-80%), η'TT is its thermodynamic expansion efficiency ' ηTT = 1 − ( Pie / Pn ) γ ' −1 γ' (3) which depends on pressure ratio Pie / Pn and GG gas specify heat ratio, γ ' . Also, cP' is the specific heat of this gas, and Tti its temperature, which is controlled through stoichiometry to values acceptable by uncooled turbines (700-1100K). Ref. 41: Turbopump System for Liquid Rocket Engines. NASA SP-8107, Aug. 1974. • Equating PP and PT yields the required mGG . Suppose now the turbine exhaust is expanded to the same exit pressure Pe as the main flow. The exhaust speed is then cGG = γ ' −1 ⎡ ⎤ 2 cP' Tt .out ⎢1 − ( Pe / Pte ) γ ⎥ ⎣ ⎦ (4) where ( ' Tt .out = Tn 1 − ηT ηTT ) (5) This speed is generally lower than the main flow exit speed, c = 2 c p Tc η ; η = 1 − (Pe / Pc ) γ −1 γ (6) where cp and η belong to the nozzle gas, and Pc , Tc , are the chamber pressure and temperature. The relative Isp loss is: − δ Isp Isp • c ⎞ mGG ⎛ ≅ • ⎜1 − GG ⎟ c ⎠ m ⎝ (7) and can be calculated once the turbine exhaust pressure Pte (and hence the turbine pressure ratio) is selected. A tradeoff is involved here: if Pte is not very low, too • much mass must be diverted (large mGG • m in Eq. 7), where as if Pte is too low, the exhaust provides almost no additional thrust (small cGG in Eq. 7). An optimum can c therefore be found. • As an example, consider the LOX-RP1 F-1 cycle, for which mox =1844 Kg/sec, • mF = 777 Kg / sec, ∆POP = 1.06 ×107 N / m2 , ∆PFP = 1.24 ×107 N / m2 , Tti = 1061 K and ηOP = 0.746, ηFP = 0.726, ηT = 0.605. Also, ρox = 1145 Kg / m3 , ρF = 810 Kg / m3 . The GG is estimated to have γ' = 1.35 and cp' = 2140 J / Kg / K , while the main gas has γ = 1.25, cP = 2080 J / Kg / K . The chamber pressure is Pc = 95 atm, and at the exit, Pe = 0.68 atm. Use of the equations above yields, after some searching, an “optimum” δI turbine pressure ratio of 23, for which sp = -0.0111. The actual F-1 engine had a Isp turbine pressure ratio of 16.4. (b) The Expander Cycle For engines, utilizing hydrogen (and possible methane) as fuel, the gas generator can be eliminated. Instead, the fuel is simply routed from the exit manifold of the nozzle cooling circuit to the turbine inlet. This is possible because hydrogen is supercritical at the pump exit, and it simply expands smoothly into an ordinary gas as it picks up heat. The resulting “Expander Cycle” is simple and efficient (the fuel is fully utilized in the thrust chamber). This cycle is used in the RL-10 engine and in the start-up sequence of the Japanese LE-5 engine (which then transitions to gasgeneratory operation). The principle limitation of this cycle is the relativity small amount of heat available from regenerative cooling, which limits applicability to chambe pressures under approximately 70 atm. A simple analysis can demonstrate this point. The pump power is given by Eq. (1) and the power derived from the fuel-driven turbine is γ F −1 ⎡ ⎤ γF P ⎛ ⎞ inj ⎢ ⎥ PT = mFηT cPF Tι i ⎢1 − ⎜ ⎟ ⎥ P ⎢⎣ ⎝ ι i ⎠ ⎥⎦ (8) where Pinj , the injector pressure, is also the turbine exhaust pressure. The turbine inlet pressure Pn , is related to the fuel pump pressure rise ∆PFP and the cooling circuit loss, ∆Pcool by ∆PFP = Pti + ∆Pcool − PTK (9) while the oxidizer pump has ∆POP = Pinj − PTK . The shaft power balance then gives (O / F ) ⎛ ρF ⎞ 1 − δTK + ⎜ ⎟ ⎝ ρox ⎠ ηOP 1 π T + δ cool − δTK =ψ ηFP γ F −1 ⎛ ⎞ ⎜1 − π T γ F ⎟ ⎜ ⎟ ⎝ ⎠ (10) where δ TK = PTK / Pinj , δ cool = ∆Pcool / Pinj , π T = Pinj Pti , and ψ = ηT ρ F cPF Tti Pinj (11) Assuming δ TK = 0.1, δcool = 0.2, ηOP = ηNF = 0.7, O / F = 5 and ρF / ρox = 69/1140 = 0.0605 , the relationship between πT and ψ , as given by (10) is shown plotted in Fig. 2. As the figure shows, the turbine inlet pressure increases rapidly when ψ drops below about 30 (at which point 1 πT = 1.33 ). In fact, as also shown in Fig. 2, the quantity ψ has a minimum value of approximately 17.76 when 1 πT = 2.68, below which no solution exists. The turbine inlet temperature Tti is 200K in the RL-10. Using also ηT = 0.7, ρF = 69 kg / m3 and CPF = 14,600 J/Kg/K, we calculate from (11) ψ ≅ 1390 / Pinj (atm) and so the maximum Pinj is 1390/17.76=78 atm. For reference, the RL-10 has PC = 32 atm, Pinj ≅ 40 atm, which corresponds to ψ = 35. Of course, as (11) indicates, higher Pinj values could be achieved if Tti could be increased further. (c) Staged Combustion Cycle For rockets where high chamber pressure as well as high efficiency is desired, the staged combustion cycle is the preferred choice. One could think of this as a modified expander cycle, in which a small amount of oxidizer is added to the fuel after the cooling circuit, thus increasing the available enthalpy for the turbine drive. As in the expander cycle, all of the propellant is entirely used in the combustion chamber. Unlike the expander, through, any oxidizer-fuel combination can be used. Two prominent examples of this cycle are the Space Shuttle Main Engine (SSME), and the Russian RD-170 booster engine (Figs. 3 and 4 from Ref. 2). In the SSME, the pre-burners are incorporated into separate fuel and oxidizer turbopump assemblies, and process most of the fuel (LH) with a small fraction of the oxidizer (LOX), producing a light “vitiated hydrogen” turbine driving gas. In the RD-170, the preburners process all of the oxidizer (LOX) and a fraction of the fuel (kerosene) to produce a fuel-lean gas which drives the single central turbine. In both cases, the turbine exhaust is ducted to the main combustor injectors, together with the remaining LOX (SSME) or kerosene (RD-170). The choice of fuel-rich preburners is precluded by carbon deposits on turbine blades and other surfaces when hydrocarbon fuel is involved. The staged combustion cycle provides the highest levels of rocket performance, but at the cost of greatly increased complexity. This is both, because of the many ducts and valves involved, and because of the very high pump exit pressures. A secondary potential difficulty, which is shared by the expander cycle, is that the turbines are unchoked and there is a possibility of low-frequency instability developing. Consider a simplified schematic (Fig. 5) of a fuel-rich cycle analogous to that of the SSME. The injector pressure, Pi, the turbine inlet temperature, Tti , and the overall O/F ratio, r, are prescribed. The heat input and pressure drop in the cooling circuit (Qcool, ∆ Pcool) are also assumed given. We wish to determine the required pressure rise ∆ PFP by the fuel pump, as well as the O/F in the pre-burners (rPB) and the fuel split SF between them. The pressure rise in the oxidizer pump is simply ∆POP = Pi − PTK . The analysis is best done iteratively. If then the turbine inlet pressure is ∆ PFP is temporarily assumed known, Pr i = PTK + ∆PFP − ∆Pcool − ∆PPB (12) where ∆PPB is the pressure drop in the pre-burner (mainly associated with injection). The required rise in the oxidizer booster pump is then ∆POBP = Pti + ∆PPB − Pi (13) The energy balance in one of the pre-burners is written as ⎛ ∆POP ∆POBP rPB ⎜ hOTK + + ρ0 ηOP ρ0 ηOBP ⎝ ⎞ r ∆PFP + Qcool + PB h f = (1 + rPB ) CPti (Tti − T ref ) ⎟ + hFTK + rst ρ F η FP ⎠ where hOTK and hFTK are the enthalpies of oxidizer and fuel in their tanks (this ignores the low pressure booster pumps) rst is the stoichiometric O/F ratio and hf the fuel heat value at the reference temperature Tref . The specific heat of the fuel-rich burnt gas is cpti. Eq. (14) can be solved for rPB if ∆PFP is assumed known. (14) The shaft power balance for the fuel turbopump is ∆PFP ρ Fη FP where η FT = S F (1 + rPB ) c pti Tn η FT is the turbine efficiency and r ' −1 ⎡ ⎤ r' ⎛ ⎞ P ⎢1 − i ⎥ ⎢ ⎝⎜ Pti ⎠⎟ ⎥ ⎢⎣ ⎦⎥ γ ' = γ ti (15) belongs to the pre-burner gas. A similar balance can be written for the oxidizer turbopump, and, by division, we can solve for the fuel split SF. ⎡ η ρ η ⎛ ∆P ∆P ⎞ ⎤ S F = ⎢1 + FT F FP ⎜ r OP + rPB OPB ⎟ ⎥ ηOPB ⎠ ⎦ ⎣ ηOT ρ0 ∆PFP ⎝ ηOP −1 (16) After rPB has been calculated from (14), SF is given by (16), and then (15) can be used as a check on the assumed ∆PFP . In reality, an outer iteration loop is required by the fact that γ ' and cpti themselves depend sensitively upon the preburner stoichiometry, rPB. With the approximations cpH2 ≅ 7.67 cal / mol / K , and c pH 2o ≅ 10.63 cal / mol / K γ '= 7.67+0.37rPB 5.68+0.37rPB (17) and then (for H2-O2), cρti = γ' R ; M’ = 2 (1+rPB) g / mol γ '− 1 M 1 (18) As an example, Fig. 3.8 shows some results in which we have used Tref = 0 K hOTK ≅ hFTK ≅ 0 and also r = 6, rst = 8, POTK =2 atm PFTK =1 atm, h f = 1.21x108 J/Kg, Tti =1100 K, η FP = 0.741 ηop =0.781, ηOBP = 0.696, η FT = 0.790, ηOT = 0.729, ∆Pcool = 0.15 Pc , ∆PPB =0.05 Pc , Qcool =7.40×105 J/kg, ∆Pi =Pi -Pc = 0.1 Pc As shown in Fig. 6, the pressure rise in the fuel pump increases more steeply than the chamber pressure, just as it did in the expander cycle. However, very high pressures are now attainable. The ∆PFP for the SSME is shown for reference. Notice how the pre-burner stoichiometry is essentially fixed by Tti , and does not vary much over the pressure range.