Illustrative nozzle exit-pressure boundary sensitivity

"""Reproduce an illustrative NASA 2000 MHD-cycle exit-pressure sensitivity. This varies a chosen exit-pressure boundary in one existing conditional cycle case. It derives area from continuity independently for each exit state, so it is not a fixed-geometry nozzle prediction or a reproduction of NASA author code. """ from __future__ import annotations import csv import json import math from pathlib import Path HERE = Path(__file__).resolve().parent INPUT = HERE / "nasa-mhd-cycle-model-results.json" Q_VALUES = (0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.4) GAMMA = 1.4 R_AIR_J_PER_KG_K = 287.05 def build_rows() -> tuple[dict, list[dict]]: cases = json.loads(INPUT.read_text(encoding="utf-8")) matches = [ row for row in cases if row.get("model_case") == "figure2-burner-cap-case" and row.get("mach") == 6.0 and row.get("extraction") == 0.25 and row.get("status") == "calculated" ] if len(matches) != 1: raise ValueError("Expected exactly one calculated Mach-6, extraction-0.25 reference row") base = matches[0] output = [] for q in Q_VALUES: p0e_over_pa = base["p0e_over_pa"] me_sq = 2 / (GAMMA - 1) * ((p0e_over_pa / q) ** ((GAMMA - 1) / GAMMA) - 1) if me_sq <= 0: raise ValueError(f"No positive exit Mach for q={q}") me = math.sqrt(me_sq) te = base["t06_K"] / (1 + (GAMMA - 1) * me_sq / 2) ue = me * math.sqrt(GAMMA * R_AIR_J_PER_KG_K * te) f = base["fuel_air_ratio"] # Continuity, assuming all air and fuel cross the same exit plane. exit_area_per_air_mass_flow = (1 + f) * R_AIR_J_PER_KG_K * te / (q * ue) momentum = (1 + f) * ue - base["freestream_speed_m_s"] pressure = exit_area_per_air_mass_flow * (q - 1) output.append({ "exit_pressure_over_ambient_q": q, "exit_mach": me, "exit_static_temperature_K": te, "exit_velocity_m_s": ue, "exit_area_over_inlet_air_mass_flow_m2_s_per_kg": exit_area_per_air_mass_flow, "momentum_specific_thrust_N_s_per_kg_air": momentum, "pressure_specific_thrust_N_s_per_kg_air": pressure, "total_specific_thrust_N_s_per_kg_air": momentum + pressure, }) return base, output def write_csv(rows: list[dict]) -> None: with (HERE / "nasa-mhd-nozzle-boundary-sensitivity.csv").open("w", newline="", encoding="utf-8") as stream: writer = csv.DictWriter(stream, fieldnames=list(rows[0])) writer.writeheader() writer.writerows(rows) def write_svg(rows: list[dict]) -> None: width, height = 1200, 800 left, right = 120, 90 x0, x1 = left, width - right q0, q1 = Q_VALUES[0], Q_VALUES[-1] px = lambda q: x0 + (q - q0) / (q1 - q0) * (x1 - x0) top0, top1 = 135, 390 bot0, bot1 = 515, 690 thrust_min, thrust_max = 650.0, 740.0 pressure_min, pressure_max = -40.0, 40.0 y_thrust = lambda value: top1 - (value - thrust_min) / (thrust_max - thrust_min) * (top1 - top0) y_pressure = lambda value: bot1 - (value - pressure_min) / (pressure_max - pressure_min) * (bot1 - bot0) colors = { "momentum_specific_thrust_N_s_per_kg_air": "#62c6ff", "pressure_specific_thrust_N_s_per_kg_air": "#ffd166", "total_specific_thrust_N_s_per_kg_air": "#ff71cf", } labels = { "momentum_specific_thrust_N_s_per_kg_air": "Momentum contribution", "pressure_specific_thrust_N_s_per_kg_air": "Pressure contribution", "total_specific_thrust_N_s_per_kg_air": "Total specific thrust", } lines = [ f'') (HERE / "nasa-mhd-nozzle-boundary-sensitivity.svg").write_text("\n".join(lines) + "\n", encoding="utf-8") def main() -> None: base, rows = build_rows() methods = { "title": "Illustrative exit-pressure boundary sensitivity for a conditional NASA/TP-2000-210387 reconstruction", "status": "Mathematical exit-state sensitivity only; not a fixed-geometry nozzle prediction, author-code reproduction, or source-validated performance result.", "source": { "record": "https://ntrs.nasa.gov/citations/20000095934", "pdf": "https://ntrs.nasa.gov/api/citations/20000095934/downloads/20000095934.pdf", "nozzle_efficiency_equations": "NASA Eqs. 54–56, PDF p. 22 (printed p. 16)", "exit_state_and_thrust_equations": "NASA Eqs. 57–64, PDF pp. 23–24 (printed pp. 17–18)", }, "reference_case": { "flight_mach": 6.0, "generator_extraction_ratio": 0.25, "source_model_case": "Figure 2 burner-cap case, not NASA author code", "gamma": GAMMA, "R_J_per_kg_K": R_AIR_J_PER_KG_K, "source_model_p0_exit_over_ambient": base["p0e_over_pa"], "source_model_T0_exit_K": base["t06_K"], "fuel_air_ratio": base["fuel_air_ratio"], "freestream_speed_m_s": base["freestream_speed_m_s"], "source_model_nozzle_total_pressure_ratio": 0.98, "source_model_accelerator_exit_temperature_K": base["t06_K"], "reported_accelerator_temperature_limit_K": 3000.0, "accelerator_temperature_cap_exceeded": base["t06_K"] > 3000.0, }, "sweep": { "exit_pressure_ratio_q_values": list(Q_VALUES), "selection": "Illustrative analyst-selected pressure boundaries, not transcribed from NASA Figure 2 and not fitted to its curve.", "held_fixed": "All upstream conditional cycle outputs, including total exit pressure, total exit temperature, fuel ratio and freestream state.", "varying_boundary": "Exit static pressure p_e / ambient static pressure p_a.", "geometry_treatment": "Exit area per inlet-air mass flow is derived separately at each q from continuity, assuming all air and fuel cross the exit plane. Therefore points do not share a fixed exit area or a fixed-geometry nozzle.", }, "equations": { "exit_mach": "M_e = sqrt(2/(gamma-1) * ((p0_e/p_e)^((gamma-1)/gamma) - 1))", "exit_temperature": "T_e = T0_e / (1 + (gamma-1)*M_e^2/2)", "exit_speed": "u_e = M_e * sqrt(gamma*R*T_e)", "continuity_area_per_air_flow": "A_e/mdot_air = (1+f)*R*T_e/(p_e*u_e)", "momentum_thrust_per_air_flow": "(1+f)*u_e-u_a", "pressure_thrust_per_air_flow": "(A_e/mdot_air)*(p_e-p_a) = (1+f)*R*T_e/u_e*(1-1/q)", "total_specific_thrust": "momentum contribution + pressure contribution", "unit_check": "Each thrust-per-mass-flow term has units m/s = N*s/kg-air.", }, "unavailable_source_inputs": [ "The Figure 2 discussion gives nozzle total-pressure ratio pi_n=0.98 but not nozzle expansion efficiency eta_s(n), the pointwise station-6/exit static-pressure relation, or a fixed exit area/geometry.", "Eqs. 54–56 therefore do not select a unique physical nozzle state from the reported Figure 2 values alone.", "This sensitivity cannot determine which q applies to NASA's plotted curve and must not be used to tune the reconstruction to that curve.", ], "limits": [ "Perfect-gas, quasi-one-dimensional reconstruction and single-stream continuity.", "All fuel is assumed to exit in the same plane; bleed, capture, spillage and installation drag are omitted.", "The reference model's accelerator-exit temperature is about 3418 K, above the report's separate 3000 K limit; this cap conflict remains.", "Near cancellation of momentum and pressure terms is a consequence of these conditional exit-state assumptions, not validation of a nozzle or vehicle.", ], "results": rows, } (HERE / "nasa-mhd-nozzle-boundary-sensitivity-methods.json").write_text(json.dumps(methods, indent=2, ensure_ascii=False) + "\n", encoding="utf-8") write_csv(rows) write_svg(rows) print(json.dumps({"rows": len(rows), "q_range": [Q_VALUES[0], Q_VALUES[-1]], "specific_thrust_range_N_s_per_kg_air": [min(x["total_specific_thrust_N_s_per_kg_air"] for x in rows), max(x["total_specific_thrust_N_s_per_kg_air"] for x in rows)], "fixed_geometry": False}, indent=2)) if __name__ == "__main__": main()