Dimensionless ionization budgets for MHD energy recycling with a reduced inlet model
DMGB | Independent researcher | Working draft v2 | 5 October 2026

Abstract
We reformulate an MHD energy ledger without fixing an arbitrary dimensional extraction value. For normalized ionization expenditure x=e/d, auxiliary demand alpha=a/d, independently evaluated useful-energy credit chi=c/d, generator conversion g and round-trip conversion r=g h s, the stipulated normalized gain is chi-1+r-(r/g)(x+alpha). A map in x and r still requires g and alpha. Normalizing total electrical loads instead by gross generation, z=(e+a)/(g d), yields the scale-independent requirement chi_required=1-r(1-z). A perfect-gas, constant-area Faraday generator followed by a normal shock and matched ideal expansion supplies a model-derived equivalent credit from local Mach, conductivity and field, conditional on gas state, length and electrical load. It is not a combusting-engine model or experimental calibration. A reference calculation returns chi=0.57460 and a negative useful-energy proxy with 60 percent electrical return. Published component metrics remain distinct. The contribution is a reproducible accounting and model-interface study; existing literature already addresses ionization affordability and inlet control, so no new physical principle is claimed.

1. Research question and objective
An affordable ionizer is necessary for a self-powered MHD bypass but is not sufficient for a net propulsive advantage. We ask what additional useful-energy benefit must be demonstrated after extraction and imperfect return. We separate electrical-bus feasibility, a reduced flow-energy proxy, and the unmodeled installed-engine objective. A numerical reduction that reproduces conservation is not by itself experimental validation.
The scope is informed by idealized cycles [1], whole-cycle pressure and cooling penalties [2], ionization-power analyses [3,4,5], and inlet control [12]. A lower-dimensional transparent audit can be useful without establishing new physics. The source matrix records overlap and retrieval limits instead of treating inaccessible full text as absent prior art.

2. Energy definitions and matching boundaries
All specific energies below use kJ per kg of inlet air. Define d >= 0 as the removed useful stream energy, g in [0,1] as the fraction reaching the electrical bus, e >= 0 as total wall-plug ionization expenditure, and a >= 0 as other electrical demands charged to that bus. Magnet, cryogenic, cooling, conditioning and sustaining loads must be assigned consistently, without omissions or double counting. These illustrative parameters are assumptions, not measured HVEPS efficiencies.
Let s in [0,1] allocate remaining bus energy to an accelerator, with h in [0,1] its electrical-to-useful-energy conversion under the chosen boundary. Energy exported elsewhere is outside the kinetic objective used here; a combined electrical/propulsive objective requires a different ledger. Heat remains in the full energy balance. Joule or generator heat is not destroyed, and any useful subsequent recovery must be included consistently. The dimensional ledger alone does not calculate that recovery; Section 4 supplies one deliberately limited ideal-expansion proxy.
The credit c denotes a separately demonstrated difference in useful energy caused by other cycle changes, evaluated against the same reference. Examples of mechanisms needing independent modeling include changed combustion completion, flow matching and nozzle recovery. It is not a free source of energy. Fuel input, captured mass, pressure thrust and installation losses must be normalized before translating it into another performance objective. An arbitrary positive c creates only a hypothetical sensitivity case.

3. Dimensionless threshold and maps
For a positive extraction d, write x=e/d, alpha=a/d, chi=c/d, and r=g h s. The electrical budget is b=g d-e-a. All returns and costs refer to the same processed inlet-air mass. Direct substitution gives
EQUATION: Delta/d = chi - 1 + r - (r/g)(x+alpha) .
This form requires g>0 and 0<=r<=g<=1. An admissible self-powered point also needs x+alpha<=g. For r>0, strict gain requires
EQUATION: x < x* = g[1+(chi-1)/r] - alpha .
An equality is break-even. If r=0, the normalized gain is chi-1 and there is no finite threshold of this form. If d=0, division by d is undefined; use the dimensional ledger. If g=0, no positive electrical load can be supplied from generation and normalization by g d is unavailable.
The round-trip product alone does not collapse the x map: identical r can arise from different generator and return efficiencies, and the electrical loads are paid after generation. We retain g as an explicit facet and alpha as a stated assumption. A scale-independent companion map instead uses z=(e+a)/(g d), the fraction of gross electrical generation spent before return:
EQUATION: Delta/d = chi - 1 + r(1-z) ; chi_required = 1-r(1-z) .
Its electrical domain is 0<=z<=1. Values of r in [0,1] require compatible components with g>=r; the map does not promise such hardware exists. Strict gain needs chi>chi_required. The transformation preserves the energy boundary and removes dimensional scale, not dependence of real efficiencies or plasma behavior on operating conditions.
FIGURE: results/universal-round-trip.png
Figure 1. Extra benefit required, normalized by extraction, versus total electrical-load fraction and round-trip conversion. Gray/hatching denotes electrical deficit. Colors are requirements, not measured or feasible engine benefits. The reference r=0.36,z=0.05 requires chi=0.658; no 400-kJ/kg value enters this map.
FIGURE: results/normalized-facets.png
Figure 2. Requested e/d-versus-round-trip maps retain generator conversion g explicitly. The three panels hold alpha=0.03. Gray means either insufficient bus energy or r>g. The masks have different physical causes; neither is a validated operating boundary in Mach, conductivity or field.

4. Reduced inlet closure for an equivalent credit
To move beyond choosing c arbitrarily, we calculate flow states in a deliberately simple inlet comparison. Both branches receive the same uniform local duct-entry state and use the same final expansion back pressure. The reference has a terminal normal shock followed by ideal expansion. The modified branch adds a uniform-section scalar-conductivity Faraday generator before that shock. This reference is not an optimized scramjet inlet: results are conditional on its large shock loss and on the omitted external-compression geometry.
Let G=rho u be constant mass flux, sigma a prescribed scalar conductivity, B a transverse applied field, K in (0,1) a constant ideal Faraday load factor, R the gas constant, gamma the specific-heat ratio and cp=gamma R/(gamma-1). With positive braking force density f=sigma u B^2(1-K), conservation gives
EQUATION: d' = f/rho ; u' = -(f/G)[gamma-K(gamma-1)]/(1-1/M^2) .
EQUATION: T' = (-K d' - u u')/cp ; h0,out = h0,in - K d .
Here primes denote derivatives with respect to axial distance ell, not time. The integral d is Lorentz mechanical work per mass; terminal electrical extraction is K d and Joule deposition is (1-K)d. Thus this model enforces g=K. Calling the stagnation-enthalpy drop d and then multiplying it by g again would double-count a loss. With S denoting specific thermodynamic entropy, the independent relation is dS/dell=(1-K)d'/T. S and ell are distinct from allocation s and normalized ionization x. The governing conservation structure is related to published MHD analyses [9], but our scalar Faraday closure is not Schneider's annular Hall device.
The solver stops at local Mach 1.05 before the sonic singularity at 1. A stopped case is not a prediction at the requested duct length and receives no final credit. The normal-shock states follow perfect-gas mass, momentum and energy conservation [13]. For either downstream state define the ideal-expansion kinetic-energy proxy
EQUATION: E = cp T0 [1-(p_back/p0)^((gamma-1)/gamma)] .
The pressure p0 here is stagnation pressure, not inlet static pressure. The expression requires p0>p_back. Nozzle efficiency is fixed to unity in both branches. With E_m and E_b denoting modified and baseline values, set
EQUATION: c_equiv = d + E_m - E_b ; chi_equiv = c_equiv/d .
This is a derived equivalent credit that makes c_equiv-d equal the independently computed inlet/expansion difference. It is not a separately measured combustion benefit. Returning a stipulated useful fraction h s of the remaining electrical bus gives Delta_proxy=E_m-E_b+h s(K d-e-a), matching the original ledger exactly. The algebraic agreement is a consistency test, not experimental corroboration.
Accordingly c is a function of local inlet Mach, sigma and B only when the gas state, interaction length, loading and back pressure are fixed. The relevant function is c_equiv(M_local,sigma,B;T_in,p_in,L,K,p_back,...). M_local is not automatically flight Mach. No conductivity-versus-ionization-cost relation is provided; prescribing sigma while sweeping e is a sensitivity study. In this closure, sigma and B enter only through the product sigma B^2: equal products are indistinguishable. No Hall-stability threshold or controllable-field limit is inferred from this scalar model. Ionizer heating, sheath drops, real-gas chemistry, wall friction/cooling, capture/bleed and the combustor remain outside the closure.

5. Numerical checks and component evidence
The illustrative reduced sweep contains 27 parameter combinations at local Mach 2, 3, 5, conductivity 1, 5, 130 S/m and fields 0, 1, 2 T; the other assumptions are T_in=800K,p_in=p_back=10000Pa,L=0.1m,K=0.60. These values define a numerical sensitivity exercise, not a proposed laboratory or flight operating envelope. One combination stops at the sonic guard and 26 reach the requested length. All zero-field cases have d=0 and therefore undefined chi; they are not assigned a fabricated normalized value.
At local Mach 3, sigma=5 S/m, B=1 T, the reduced model gives d=7.784959kJ/kg, electrical extraction 4.670975kJ/kg and c_equiv=4.473258kJ/kg, hence chi_equiv=0.574603. If all electricity is returned at h=0.60 and both ionization and other electrical costs are set to zero, r=0.36 and break-even requires chi=0.64. The proxy difference is -0.509116kJ/kg. This is a negative result for the declared crude comparison, not a universal engine limit or a calibrated thrust prediction. Charging positive electrical costs without state changes makes this slice worse; modeling plasma heat/state changes would require another closure.
For continuity, the original teaching slice d=400kJ/kg,g=h=0.60,s=1,a=12kJ/kg becomes r=0.36,alpha=0.03,x=0,z=0.05. Its 263.2kJ/kg requirement is exactly chi=0.658. Scaling d,e,a,c together leaves that normalized requirement unchanged. The inherited 81 extraction slices remain in the supplement as bookkeeping tests, not experimental cases.
The released scripts include exact-rational checks, dimensionless scaling and electrical-domain checks, zero-field and shock-conservation checks, and independent numerical review of the reduced inlet. Solver agreement verifies the stipulated equations only. Source-number arithmetic remains separate from flow simulation, and no statistical confidence interval is assigned to assumed parameters.
HVEPS Test 3 [6] reports 57.82 A and 1.95 ohm at the external load. Independent arithmetic gives I^2 R = 6.51914718 kW, agreeing with 6.52 kW after rounding. The same report distinguishes 14.38 kW estimated active-channel output and 14.80 kW calculated matched-load maximum. Their difference does not identify a generator efficiency. A complete synchronized input-energy and actual mass-flow denominator is needed. We therefore use an electrical-boundary benchmark, not a fitted generator efficiency.
The Ames comparison [7] reports approximate velocity increases of 27 and 38 percent for Runs 15 and 16. Published selected two-/three-dimensional results are 26/26 and 31/30 percent. Conductivities used are 130/140 S/m. These are component comparisons with inferred velocity, sheath uncertainty and field selection. Re-entering 1.27 and 1.38 reproduces source ratios, not an independent flow prediction. Squaring an averaged velocity is not a mass-flux-weighted kinetic-energy measurement. The earlier core-flow load-factor interpretation [8] excludes some device losses; its 43.48-47.62 percent kinetic fraction cannot directly replace an isentropic efficiency [9].
Consequently these publications cannot calibrate an integrated bypass on two points. A predictive validation would require raw profiles and electrical traces, documented diagnostic models, a shared energy boundary, uncertainty and held-out cases. These remain absent from this reduced calculation. The new reduced inlet supplies a conditional local-flow dependence, but it is neither fitted to these percentages nor calibrated as a whole engine.

6. Conductivity evidence and the next measurements
The 827/151 ratio in [3] is 5.4768 and the difference is 676 MW. Both are rounded modeled powers in one seeded-air case, including ideal ionizer efficiency. The table's 7.18 kg/s is fuel flow; dividing by it would not give energy per kg of inlet air. It is therefore not used to normalize our illustrative map. This case is not a universal bound on other ionization mechanisms.
Schneider [9] compares required fractions 1.90e-6 and 9.52e-6 for 1 and 5 S/m with cited discharge estimates 1.2e-8. Ratios are approximately 158 and 793 under that report's assumptions. This historical comparison is narrower than saying no intermediate-conductivity measurements exist. A 2023 study [5] considers ideal extraction/ionization ratios near 24-30 in a particular CO2 calculation; its experimental comparison is qualitative. Those results cannot be transferred directly to the air case above or treated as an integrated-loop measurement.
The defensible missing evidence is a matched, uncertainty-qualified relation among effective conductivity, wall-plug ionization cost per processed air mass, and generator/accelerator performance under shared conditions. A gap between selected studies is not proof of a universal empty conductivity interval. Additional primary literature retrieval, including conductivity diagnostics [10,11], is part of the remaining novelty and evidence review.

This is a measurement specification for a qualified plasma/flow laboratory, not a hardware construction or operating procedure. Pre-register a shared composition and thermodynamic state and target coverage in bands 1-5, 5-20, 20-50 and 50-130 S/m, without assuming those states are attainable. Define whether sigma denotes local, tensor, spatially averaged or effective channel conductivity. Track residence-time and discharge-duty distributions and distinguish electrode/sheath impedance from bulk response.
Record synchronized inlet mass flow, gas state/composition, velocity profiles, field map, wall-plug and delivered ionizer energy, generator terminal output, accelerator input, auxiliaries and wall heat. Integrate e = integral(P_ion_wall dt)/integral(mass_flow dt) over the same valid window. Keep stored-energy transients and startup costs separate from steady operation; report both when relevant.
Use independent conductivity and velocity diagnostics with stated calibration covariance. Compare matched flow-only, plasma-only, field-only and combined controls, randomized order and repeat runs. Choose repeat count from a pilot variance and a preregistered interval-width target, rather than inventing an adequate sample size now. Publish raw traces, calibration, saturation flags, rejected-run rules and uncertainty propagation. For measured b, d and conversion k=s h, the gain sensitivity to (c,d,k,b) is (1,-1,b,k); use the full covariance matrix, not an independence assumption if channels share diagnostics. The resulting J Sigma J-transpose is a first-order approximation; broad or asymmetric uncertainties require nonlinear joint propagation.
Component milestones are conductivity attainment, positive bus balance, and useful flow-energy response. A later integrated comparison must measure momentum and pressure force against the same reference and fuel input. A confidence interval crossing zero is inconclusive. Successful component tests do not establish installed vehicle benefit.

7. Prior art and limits on the contribution
The version 2 evidence matrix contains ten primary-led entries and a dated query log, distinguishing complete or targeted text inspection, abstracts and unresolved journal full text. Existing papers already couple ionization power, electromagnetic flow interaction and inlet behavior. A later reduced-order screening paper is also identified as a close comparison requiring full examination. Dimensionless rewriting, energy affordability and magnetic inlet control must not be advertised as new physical concepts.
Our narrower candidate contribution is an openly reproducible interface between a boundary-explicit energy budget and a deliberately limited inlet-quality model, with checks against incompatible efficiency definitions. Demonstrating an unreported robust limit would require additional matched validation and a defensible reference inlet. The completed targeted screen does not establish originality and is not represented as an exhaustive full-text review. Its exact remaining retrieval items are supplied with the supplement.
The model has no plasma kinetics or conductivity-cost closure, no combustion and no installed-aircraft accounting. A sonic cutoff is a numerical validity guard, not a controllability criterion. Grid counts do not establish experimental credibility. A positive or negative value from the chosen normal-shock reference cannot be generalized to a well-designed oblique-shock or shock-train inlet.

8. Authorship, reproducibility and publication status
The working manuscript uses the project's existing author identity DMGB and affiliation Independent researcher. Author metadata no longer blocks calculations or writing. Actual arXiv submission still requires the author's account, final review, distribution-license choice and any applicable endorsement. No submission or external outreach has occurred.
The supplement includes the editable text, PDF producer, numerical scripts, CSVs and source-access matrix. Existing publications are cited, not copied into the distribution. Exact rational arithmetic checks prevent implementation roundoff from masquerading as new precision; changing numeral base does not improve experimental information. This remains a research working draft, not an accepted preprint or peer-reviewed result.

References
[1] Litchford et al. Thermodynamic Cycle Analysis of Magnetohydrodynamic-Bypass Hypersonic Airbreathing Engines. NASA/TP-2000-210387. https://ntrs.nasa.gov/citations/20000095934
[2] Riggins, D. W. Analysis of the Magnetohydrodynamic Energy Bypass Engine for High-Speed Airbreathing Propulsion. Journal of Propulsion and Power (2004). https://doi.org/10.2514/1.4980
[3] Park, C., Bogdanoff, D. W., and Mehta, U. Power Requirement for Nonequilibrium MHD-Bypass Scramjet. NASA NTRS 20010066491, Table 1. https://ntrs.nasa.gov/citations/20010066491
[4] Macheret, S. O., Shneider, M. N., and Miles, R. B. Magnetohydrodynamic Control of Hypersonic Flows and Scramjet Inlets Using Electron Beam Ionization. AIAA Journal 40, 74-81 (2002). https://doi.org/10.2514/2.1616
[5] Starikovskiy, A. Y., Aleksandrov, N. L., and Shneider, M. N. Analysis of the efficiency of MHD cycle supported by nanosecond pulsed discharge pre-ionization. Plasma Sources Science and Technology 32, 035005 (2023), sections 6-7. https://doi.org/10.1088/1361-6595/acc053
[6] Lineberry et al. HVEPS Scramjet-Driven MHD Power Demonstration Test Results. AIAA 2007-3880; AFRL-PR-WP-TP-2007-233, Table VI, printed p.27. https://doi.org/10.2514/6.2007-3880
[7] Kato, Tannehill, Gupta and Mehta. Numerical Simulation of 3-D Supersonic Viscous Flow in an Experimental MHD Channel. AIAA 2004-0317, printed pp.5-6. https://ntrs.nasa.gov/citations/20040105526
[8] Bogdanoff, D. W., and Mehta, U. Experimental Demonstration of MHD Acceleration. AIAA 2003-4285, section IV.E, printed p.19. https://doi.org/10.2514/6.2003-4285
[9] Schneider, S. J. Annular MHD Physics for Turbojet Energy Bypass. NASA/TM-2011-217210, printed pp.8-11. https://ntrs.nasa.gov/citations/20110016528
[10] Murray et al. Magnetohydrodynamic power generation using externally ionized cold supersonic air as working fluid. AIAA Journal (2006). https://doi.org/10.2514/1.11611 (Author-hosted full text inspected for targeted prior-art comparison; no calibration fit used.)
[11] Measurement of Flow Conductivity and Density Fluctuations in Supersonic Nonequilibrium Magnetohydrodynamic Flows. https://doi.org/10.2514/1.13736 (Retrieval lead; no numerical result used here.)
[12] Shneider, Macheret and Miles. Analysis of Magnetohydrodynamic Control of Scramjet Inlets. https://doi.org/10.2514/1.3998
[13] NASA Glenn. Normal Shock Wave Equations, derived from NACA 1135. https://www.grc.nasa.gov/www/k-12/airplane/normal.html
