Hydrowheel Buoyancy Engine — Technical Overview
System: Buoyancy + hydrofoil rotational energy converter Status: v1.1 complete. Net positive energy confirmed. COP_max = 1.210, below the original 1.5 stretch target. Analysis method: Phased first-principles derivation with explicit validation gates at each stage (see Validation section). No CFD; XFOIL + analytical methods, appropriate for a feasibility study of this scope.
1. System Description
30 open-bottom cylindrical vessels (0.457 m diameter × 1.219 m length, 0.2002 m³ each) arranged on 3 vertical loops orbiting a central shaft, inside a fresh-water cylinder 7.32 m diameter × 18.29 m deep.
Ascending vessels are air-filled at the bottom of the loop and rise by buoyancy. Descending vessels are water-filled and pulled down by chain coupling to the ascending side. Every vessel carries a hydrofoil (NACA 0012, AR = 4). Foils are tacked (flip angle of attack) so both ascending and descending vessels contribute torque in the same rotational direction. The enclosed water body co-rotates with the vessel loops, which reduces horizontal drag on the vessels without reducing the vertical velocity the foils depend on for lift.
Air is injected over a fill window of 1/4 loop circumference.
Key parameters
Parameter
Symbol
Value
Vessel diameter
d
0.457 m (18 in)
Vessel length
L
1.219 m (4 ft)
Vessel volume
V₀
0.2002 m³ (7.069 ft³)
Depth (bottom to surface)
H
18.288 m (60 ft)
Pressure at depth
P_d
2.7669 atm (280,500 Pa)
Vessel count
N
30 (3 loops × 10)
Cylinder diameter
D
7.32 m (24 ft)
Foil profile
—
NACA 0012, AR = 4
Water density (fresh, 20°C)
ρ_w
998.2 kg/m³
2. Notation
Symbol
Quantity
Units
P(z)
Absolute hydrostatic pressure at height z
Pa
V(z)
Air volume in vessel at height z (Boyle's law)
m³
z
Height above tank bottom (z=0 at bottom, z=H at surface)
m
W_iso
Isothermal compression work
J
W_adia
Adiabatic (isentropic) compression work
J
W_pump
Actual pump work = W_adia / η_c
J
W_buoy
Buoyancy work during ascent
J
η_c
Isentropic compressor efficiency
— (0.65–0.85)
COP
System coefficient of performance = W_shaft_out / W_pump_in
—
L/D
Lift-to-drag ratio (force ratio, not power ratio)
—
C_L, C_D
Lift and drag coefficients
—
α (AoA)
Hydrofoil angle of attack, relative to flow direction
degrees
v_v, v_h
Vessel vertical / horizontal velocity components
m/s
f
Co-rotation fraction (0 = none, 1 = full solid-body)
—
P_corot
Co-rotation maintenance power (wall friction)
W
λ
Ratio parameter for rotating-arm force sweep
—
Sign convention: z increases upward; P(z) decreases with z; V(z) increases with z (Boyle's law); F_vert is negative when it opposes vessel ascent.
3. Governing Physics
3.1 Compression work
Isothermal (theoretical minimum):
W_iso = P_atm × V₀ × ln(P_d / P_atm)
      = 101,325 × 0.2002 × ln(2.7669)
      = 20,644.6 J per vessel cycle
Adiabatic (γ = 1.4):
W_adia = [γ/(γ−1)] × P_atm × V₀ × [(P_d/P_atm)^((γ−1)/γ) − 1]
       = 23,959.5 J per vessel cycle
Real pump work: W_pump = W_adia / η_c → 28,188–36,861 J at η_c = 0.85–0.65.
3.2 Buoyancy work (thermodynamic identity)
F_b(z) = ρ_w × g × V(z),  V(z) = V₀ × P_atm / P(z)
W_buoy = ∫₀ᴴ F_b(z) dz = P_atm × V₀ × ln(P_d/P_atm) = W_iso
This identity was confirmed numerically to 2×10⁻⁷% relative error. Buoyancy alone is energy-neutral at best (COP_ideal_max = 0.73 once real losses are included); it cannot be counted as net output. Using the constant-volume approximation (F_b × H) instead of the variable-volume integral overestimates buoyancy work by ~74% and is the single most common error in this class of analysis.
3.3 Hydrofoil lift, drag, and torque
2D section data (NACA TR-824, NACA 0012, Re ~ 10⁶):
α
C_L
C_D
L/D
5°
0.55
0.008
69
8°
0.86
0.011
78
10°
1.06
0.013
82
12°
1.20
0.016
75
Finite-span correction (Prandtl lifting line, AR = 4): C_L,3D = C_L,2D / (1 + 2/AR), C_D,induced = C_L,3D² / (π e AR), e ≈ 0.85–0.95. Effective L/D at AR = 4 is roughly 15–35, well below the 2D values above.
Critical distinction: L/D is a force ratio, not a power ratio. Power extracted depends on the velocity triangle:
P_lift = F_L × v_h        (useful torque contribution)
P_drag = F_D × |v|         (loss)
Net foil power = F_D × |v| × [(L/D)(v_h/|v|) − 1]
Net positive requires L/D > |v|/v_h. Treating L/D directly as an efficiency multiplier is a documented pitfall (see Section 5).
3.4 Co-rotation
Modeled parametrically as fraction f of full solid-body rotation (f=0: no co-rotation, f=1: water and vessels rotate together, zero relative horizontal velocity). Reduces horizontal drag without affecting the vertical velocity the foils depend on.
Wall-friction maintenance cost (Taylor-Couette / turbulent flat plate estimate):
τ_w = ½ × ρ_w × C_f × (ωR)²,   C_f ≈ 0.074 × Re_wall^(−0.2)
P_corot = τ_w × 2πRL × ωR
At ωR = 1 m/s, R = 3.66 m: P_corot ≈ 1.3 kW (order-of-magnitude estimate). At the system's stall-limited co-rotation fraction (f_stall = 0.294), full-scale results show P_corot = 22.2 kW against 46.8 kW of drag savings, a net positive of roughly 24.6 kW before the v³ velocity correction in Section 4.
4. Results by Milestone
v1.0 Feasibility Study (Phases 1–4, shipped 2026-03-19)
Compression work bounds confirmed: W_iso = 20,644.6 J, W_adia = 23,959.5 J per vessel cycle
Buoyancy identity gate PASSED (W_buoy = W_iso to 2×10⁻⁷% relative error)
Fill feasibility: GO. 147–295 SCFM at 26 psig across all velocity cases, achievable with a medium industrial compressor
Rotating-arm NACA 0012 force sweep: F_tangential > 0 for λ ∈ [0.3, 1.27]. Ascending-only COP_partial = 1.21 at design point
Descending tacking confirmed by explicit vector geometry (Darrieus rotor analogy). Combined 24-vessel COP_partial = 2.06 at λ = 0.9 (upper bound, pre-correction)
Co-rotation: f_stall (angular-momentum, stall-limited) = 0.294. Net positive: 46.8 kW drag savings vs. 0.72 kW maintenance cost at nominal velocity
Critical correction: F_vert (vertical force component from the foil, opposing ascent) was found to reduce the self-consistent loop velocity from 3.714 m/s to 2.384 m/s, a 36% drop. This is a fundamental kinematic effect (lift acts perpendicular to relative velocity) — it is not fixable by reversing foil mounting orientation.
Co-rotation benefit rescales with v³, dropping from 46.8 kW to ~12.4 kW net after the velocity correction
v1.0 verdict: NO_GO against the 1.5 target. Corrected COP_nominal = 0.925 (range 0.811–1.186 across 9 η_c × loss scenarios)
v1.1 AoA Parametric Sweep (Phases 5–6, shipped 2026-03-21)
Full 16-point angle-of-attack sweep (1°–15°) using a brentq solver anchored to the v1.0 result (reproduced Phase 4 anchor to within 0.001%)
AoA_optimal = 2.0°: co-rotation gain (+168 kJ) narrowly outpaces foil-torque loss (−147 kJ) relative to the 10° baseline
COP_max = 1.210 at (η_c = 0.85, loss_frac = 5%) — this is the ceiling for the current geometry
Required η_c for a 1.5 target: η_c* = 1.054, which exceeds the isothermal compression limit (η_c ≤ 1.0) — physically unreachable at this depth and geometry
Even at ideal compression (η_c = 1.0, 5% losses): COP_max = 1.423, still short of 1.5
AoA_optimal confirmed scenario-independent across all 9 η_c × loss combinations (W_gross(AoA) is the only AoA-dependent COP term)
v1.2 Purge Thrust and Tail Foil (Phases 7–8, in progress)
Testing two previously excluded contributions:
Purge jet thrust: reaction force from water ejected during isothermal air expansion in an open-bottom vessel. Must be checked against the W_buoy = W_iso identity to confirm it isn't double-counted energy already inside the buoyancy integral.
Tail foil: a second foil (span 0.457 m, chord ≤ 0.457 m) positioned to catch the purge jet's exit velocity.
Target: revised 9-scenario COP table with margin above the COP = 1.0 net-positive threshold.
v1.3 Differential Rotation Analysis (planned)
Tests whether water rotating faster than the vessel arms (speed ratio r = v_water/v_arm, 1.0–1.5) acts as a COP multiplier, additive boost, or stall trigger, via a shifted apparent flow vector at the foil. Energy cost of maintaining faster-than-arm rotation is explicitly out of scope for this pass (treated as free/externally driven, e.g. by wave action).
5. Validation Gates and Known Pitfalls
Six critical pitfalls were identified and guarded against by explicit numerical checks at each phase:
#
Pitfall
Guard
C1
Treating buoyancy work as net output (double-counts pump energy)
W_buoy = W_iso gate, < 1% tolerance, checked every phase
C2
Treating L/D as a power ratio instead of a force ratio
Explicit velocity-triangle power balance (P = F·v) required
C3
Claiming co-rotation drag reduction without its maintenance cost
P_corot computed and included in every energy balance
C4
Treating ascending and descending torque as independently additive
Chain tension tracked as the coupling variable between loop halves
C5
Applying the heat-pump COP>1 analogy without an external energy source
External reservoir must be identified explicitly before claiming η > 1.0
C6
Double-counting purge jet thrust on top of the buoyancy integral
Being resolved explicitly in v1.2 (PROP-01)
Mandatory numerical gates, all passed to date:
W_buoyancy = W_isothermal to < 1% (achieved: 2×10⁻⁷%)
Lossless-system COP diagnostic = 1.0 ± 0.01% (achieved) — note the operating system is a net energy producer once hydrofoil torque is included, so the operating-point lossless gate is COP_lossless = 2.20, not 1.0; the 1.0 gate applies to the buoyancy-only sub-system
NACA 0012 benchmark, Re=10⁶, α=0°: C_D ≈ 0.006 ± 15% (XFOIL calibration check)
6. Computational Approach
Python 3.10+ (numpy, scipy, matplotlib) for all energy balance, integration, and parameter sweeps. XFOIL 6.99 for 2D hydrofoil section polars (validated against NACA TR-824 / Ladson et al. 1988). Prandtl lifting-line (closed-form, 3 lines of numpy) for finite-span correction. Full 3D CFD was explicitly evaluated and rejected as disproportionate to a feasibility-level analysis: it adds days of runtime per configuration without changing the integrated energy quantities that determine the go/no-go verdict.
Parameter sweep: 30 L/D values × 5 η_c values × 20 co-rotation fractions = 3,000 evaluations, sub-second on a standard desktop.
7. Open Questions
Does the complete system balance (with purge thrust and tail foil) push COP meaningfully above 1.21?
Does differential rotation (v1.3) act as a genuine multiplier or trigger foil stall?
What co-rotation fraction is achievable in steady state given the discrete (not continuous) vessel motion pattern?
Tack-flip mechanism energy loss is not yet modeled analytically — flagged as the highest-priority measurement to take from a physical prototype
Path to COP ≥ 1.5 (if desired) requires a design change, not a further efficiency optimization: greater depth, more/larger vessels, a fundamentally different pressure ratio per cycle, or a genuinely external energy input such as sustained wave coupling
8. Key References
Abbott, I.H. & von Doenhoff, A.E. (1959). Theory of Wing Sections. Dover.
NACA TR-824 (Abbott et al., 1945). NASA NTRS.
Çengel, Y. & Boles, M. Thermodynamics: An Engineering Approach, 9th ed.
Hoerner, S.F. (1965). Fluid-Dynamic Drag.
Schlichting, H. & Gersten, K. Boundary Layer Theory, 9th ed.
Greenspan, H.P. & Howard, L.N. (1963). J. Fluid Mech. 17(3), 385–404.
Webb, D.C., Simonetti, P.J., & Jones, C.P. (2001). IEEE J. Ocean. Eng. 26(4), 447–452. (Slocum glider, closest operational analogue)
Pimm, A.J. et al. (2012). Energy 41(1). (Underwater CAES thermodynamics)
Compiled from GPD (Get Physics Done) analysis, phases 1–6, March 2026. Phases 7 onward (v1.2, v1.3) in progress; this document will need an update once those close.