16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 30: Dynamics of Turbopump Systems: The Shuttle Engine

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1 6.5, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 30: Dynaics of Turbopup Systes: The Shuttle Engine Dynaics of the Space Shuttle Main Engine Oxidizer Pressurization Subsystes Selected Sub-Model In the coplete SSME engine, all variables affect each other in coplex ways. In order to test our fault detection algoriths, a dynaic subsyste is desired, with reduced order, but with the unodelled states interacting as weakly as possible with those odelled. Attention was focused on the liquid oxygen subsyste for two ain reasons: (a) The O/F ratio at rated power is 6, so that the LOX dynaics should doinate over the LH effects wherever they interact, and (b) The turbopup pre-burners are run very fuel-rich in order to liit the turbine inlet teperatures below the etallurgical liits of the uncooled blades; sall excursions of the LOX flow to the pre-burners are then iediately translated into large and potentially critical turbine teperature excursions. In our subodel we therefore focus attention on the LOX turbopup, which feeds both, the ain LOX injectors, and (after the boost stage) the two turbine preburners. Also odeled are the dynaics of the LOX feeding line to the pre-burners, as well as to the ain LOX valve and ain injector, plus the LOX pre-burner itself and the ain chaber pressure. Variations of the LH-related states should indeed couple weakly to the LOX syste, ainly through LH flow variations onto the preburners (insensitive, since they are fuel-rich), and into the ain chaber (choked flow, no feedback). The Dynaic Equations There are three types of dynaic equations to be considered: () Rotational dynaics of the LOX turbopups. () Equations expressing the liquid inertia under pressure difference variations (analogous to inductance in electric circuits). (3) Equations expressing the ability of cavities to store fluid due to its copressibility under pressure fluctuations (analogous to capacitive effects). () Rotational Dynaics If I OTP is the oent of the inertia of the Oxidizer Turbo Pup (OTP) rotor, angular velocity, τ OT the torque delivered by the OTP turbine, τ OP the torque absorbed by the ain oxidizer pup stage, and τ OP 3 the torque absorbed by the oxidizer booster pup, then 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page of Ω 0 its

2 dω = τ τ τ () dt 0 OTP OT OP OP3 I In a hybrid syste where Ω 0 is in rad/sec, t in seconds, and the torque in lbin, the constant I OTP has a value 0.96 (which iplies I OTP = 4 lb in ). 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page of

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4 () Inertia of LOX in pre-burner coon supply line This is a prototypical equation of type (), and, in order to illustrate the underlying physics, we well give here a brief derivation of it. Consider a pipe of length L and cross-section A, fed by the booster pup discharge at a pressure P OD3, and having a ean pressure P POS (Pre-burner Oxidizer Supply). Frictional forces along the pipe, and at bends and restrictions, contribute a total pressure drop λ ρυ, where ρ and υ are the LOX density and velocity, and λ (of order unity) is a pressure loss coefficient. The liquid is then acted on in the forward direction by a net force POD3 λ ρυ PPOS A.. The ass of liquid in the pipe is ρ AL, and we ust have dυ λ ρal = P P ρυ POS A dt ( ) OD3 i Now, the flow rate in the pipe is OP 3 = ρ υ A, so the equation can be re-written as i dop 3 λ L = P 3 3 OP OP PPOS dt ρ A A or L dop 3 A dt OP3 = P P K POS OP 3 () where K = λ ρ A (3) In units of lb / sec for and lb / in for P, the constants have the values L =, K = A 00 in and λ = 0.06 ( ) A in. A This iplies L = 3.86 (3)Fluid Capacitance in preburner LOX supply line below: This is a prototypical equation of type (3), and we also provide a derivation Considering again the POS supply line, it receives LOX flow at a rate OP 3 fro the booster pup, and discharges FPO into the fuel preburner (FP) and into OPO the oxidizer pre-burner (OP), plus a sall aount which is diverted to cool the pup. Under dynaic conditions, there is a (generally non-zero) net inflow 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page 4 of OP C

5 3. Let ρ V be the ass stored in the pipe, where OP FPO OPO OP C V π D = 4 L is the volue available. Then we ust have d ( ρ V) dt = (4) Even though LOX is a liquid, it has finite copressibility at the very high pressures involved here. This is easured by the therodynaic paraeter K dρ 0 6 = 5 0 Pa in / lb ρ dp In general, the volue V also varies slightly under pressure fluctuations, but it can be shown that this effect is secondary. We therefore rewrite (4) as d ρ dp dp dt ρ POS ( ρv) = 3 POS LOX OP FPO OPO OP C (5) In the sae units as before, the factor dρ ( ρv ) has a value of POS ρ dp LOX. Using dρ in / lb, this iplies a line volue V = 409 in, which 38,0 ρ dp cobined with previously estiated L / A = 3.86 in, yields L 40 in, A 0 in. These are not expected to be exact diensions of the POS line, because the odel lups together several secondary inertios and capacitances, but they do appear reasonable. Incidentally, fro the previous result λ 0.06 A, we now estiate λ.8, again a reasonable value for a pressure loss factor. The Coplete Subodel In addition to the three equations derived above, there are three others of the fluid inertia type and three others of the fluid capacitance type. The coplete subodel, in the sae unit used so far, is shown in Table.on next page. Equations (6), (7), (8) are the ones just derived. Eq. (9) deals with the inertia of the LOX oving through the valve and the injectors of the Fuel Preburner (FP) under the fluctuating drive of the pressure difference P P, less the pressure drops in the valve and in the injectors. These drops have the characteristic for, just as in Eq. (), but, in addition the valve open area fraction A/ A FPV appear squared in the denoinator, as it should according to Eq. (3). This area fraction will act as one of our control variables. Eq. (0) is identical in structure to Eq. (9), but refers to the fluid inertia in the LOX doe of the Oxidizer Preburner (OP). Once again, the OP valve area fraction A/ A OPV appears here as a control variable. 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page 5 of POS FP ( in)

6 The reaining inertia-type equation is Eq. (3), which refers to the LOX oving through the Main Oxidizer Valve (MOV) under the drive of the difference between the ain oxidizer discharge pressure,, and the ain cobustor pressure, P c, less the su of the pressure drops in the MOV (assued 00% open) and the injectors. The reaining three capacitance-type equations are Eqs. (), () and (4). Eq. () describes accuulation of gas in the Oxidizer Preburner (OP), with ass flow OPF (the un-odelled fuel flow into the OP) plus P OD OPO (the LOX flow into the OP) entering, and alost all of the gas flow input to the oxidizer turbine, OT, leaving. Table: DYNAMICS OF LOX PRESSURIZATION SUBSYSTEM Equation No. Equation Description Tie Constant (sec.) (6) dω Rotational O = τot τop τop3 dynaics of 0.96 dt OTP (7) din OP3 LOX inertia in = POD3 PPOS OP3 preburner 00 dt supply line (8) dp POS Mass storage = OP 3 FPO OPO OP C in preburner 380 dt supply line (9) i LOX inertia in d FPO = FPO injector doe PPOS PFP OPO dt A / AFPV of FP (0) din OPO 0.60 OPO = P P.463 POS OP dt A / AOPV () () 0, dp dp OP dt F dt = OPF OPO OT + = + M.085 FT OT FT OT (3) d MOV = POD PC MOV 5 dt (4) dp c = F + MOV CN 4000 dt OPO LOX inertia in injector doe of OP Mass storage in Oxidizer Preburner Mass storage in fuel ducts to injector LOX inertia in ain injector doe Mass storage in ain cobustor , Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page 6 of

7 Equation () describes the gas accuulation in the two large ducts which bring the partially oxidized hydrogen to the ain chaber injector doe. Feeding this volue are the (unodelled) discharge flows on the ain Fuel Turbine and of FT the low pressure Fuel Turbine OT FT, plus the discharge of the ain Oxidizer Turbine, ; leaving this volue is basically the ain Fuel Injector flow, plus soe FI saller LH cooling flows, up to.085. FI Finally, Eq. (4) governs the changes in the ain cobustor pressure, P c, due to ass accuulation. The ass inputs are the fuel and oxidizer injector flows (, ), while the ass loss is the nozzle flow rate. FI MOV CN Characteristic Ties For each of the dynaic equations (6)-(4), we can estiate the characteristic tie constant, which provides sae preliinary appreciation for the dynaics of the syste. For this purpose, we balance the rate ter with one of the doinant ters on the right; for instance, for Eq. (6), the tie constant is Ω τ = O, with and evaluated at their noinal values (rated power). These 0.9τ OT Ω O τ OT tie constraints are included in Table. The preburner supply line flow and the ain cobustor pressure are seen to adjust rapidly (under sec). Filling and eptying of the Oxidizer Preburner is relatively slow (4 sec), and the shaft speed of the OTP is very slow (58 sec). All other dynaics are coparable in speed with tie constants of a few sec. Calculation of non-state variables The sequence of algebraic coputations (no additional dynaics) required to calculate the right-hand-sides of Eqs. (6)-(4) is suarized in Appendix. The data for these calculations are the values of the nine state variables, the values of the control variables (preburner valve openings), and a few unodelled variables arising fro the fuel side of the overall syste. The latter are generally kept at their noinal values. 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page 7 of

8 Appendix B. Steady State Values fro the SSME Therodynaic Model 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page 8 of

9 Appendix C. State Variables in a siulated Throttling Sequence 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page 9 of

10 Appendix D. Characteristic Ties (approx.) for the SSME Dynaic Model 6.5, Rocket Propulsion Lecture 30 Prof. Manuel Martinez-Sanchez Page 0 of

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