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Hydraulic Power Budget: Complete Calculation at Every Stage

COMPLETE POWER BUDGET CALCULATION

STAGE 1: AVAILABLE HYDRAULIC POWER

P = rho g Q h = 1000 9.81 Q 47 = 461 * Q watts (Q in litres/second)

Q = 1 L/s (garden hose) → P = 461 W
Q = 10 L/s (large faucet) → P = 4,610 W
Q = 100 L/s (fire hose) → P = 46,100 W
Q = 372 L/s (full pipe 0.3m/s)→ P = 171,500 W

STAGE 2: WATER HAMMER PRESSURE

delta_p = rho c delta_v = 1,480,000 * v pascals

v = 0.3m/s → delta_p = 444,000 Pa = 4.4 atm
v = 0.5m/s → delta_p = 740,000 Pa = 7.3 atm
v = 1.0m/s → delta_p = 1,480,000 Pa = 14.6 atm
v = 2.0m/s → delta_p = 2,960,000 Pa = 29.2 atm

Structural check: max hammer (2.96 MPa) vs limestone strength (20-170 MPa) → safety factor 7x-57x. Stone survives. PROVEN.

STAGE 3: RAM PUMP ACOUSTIC EFFICIENCY

Hydraulic ram pumps convert 60-80% of kinetic energy to pressure wave energy. Using conservative 60%:

Q = 10 L/s: P_acoustic = 0.60 * 4,610 = 2,766 W
Q = 100 L/s: P_acoustic = 0.60 * 46,100 = 27,660 W
Q = 372 L/s: P_acoustic = 0.60 * 171,500 = 102,900 W

STAGE 4: WATER-TO-STONE TRANSMISSION

T = 4 Z_water Z_limestone / (Z_water + Z_limestone)^2

= 4 1.48e6 8.75e6 / (1.48e6 + 8.75e6)^2
= 5.18e13 / 1.047e14
= 0.495 = 49.5%

Half the acoustic energy transmits from water to stone. This is excellent coupling — far better than air-to-stone.

STAGE 5: ATTENUATION THROUGH 70m BEDROCK

Limestone attenuation at 5-50Hz: 0.01-0.05dB/m Total through 70m: 0.7-3.5dB Power factor: 10^(-3.5/10) to 10^(-0.7/10) = 0.45 to 0.85 Average: 0.65 (65% transmission)

Combined water→stone→KC transmission: 0.495 * 0.65 = 0.32 = 32%

One-third of water hammer acoustic energy reaches the King's Chamber granite through the bedrock.

STAGE 6: KING'S CHAMBER CONDITIONS

CONSERVATIVE (Q=10 L/s, P_hydraulic=4,610 W):

P_acoustic = 2,766 W

P_KC (32%) = 885 W

Intensity on ceiling (54.8m^2): I = 16.1 W/m^2
Pressure amplitude in granite:
p = sqrt(2 I Z_granite)
= sqrt(2 16.1 1.35e7) = 20,850 Pa = 0.21 atm
With resonance Q=500:
p_resonant = p sqrt(Q) = 20,850 22.4 = 466,000 Pa
= 4.6 atmospheres oscillating in the granite

MODERATE (Q=100 L/s, P_hydraulic=46,100 W):

P_KC = 8,870 W

Pressure (raw): 66,000 Pa
Pressure (Q=500): 1,476,000 Pa = 14.6 atm

FULL PIPE (Q=372 L/s, P_hydraulic=171,500 W):

P_KC = 32,930 W

Pressure (raw): 127,300 Pa = 1.26 atm
Pressure (Q=500): 2,847,000 Pa = 28 atm

Granite fatigue check at 2.85 MPa oscillating:

Granite fatigue limit (~10^7 cycles): 52-168 MPa
Safety factor: 18x-59x. Granite survives indefinitely.

STAGE 7: PIEZOELECTRIC GENERATION IN GRANITE

Granite = ~25% quartz (piezoelectric). Random orientation reduces effective coefficient. g_eff ~ 3.75e-3 V*m/N.

VOLTAGE per granite beam:

V = g_eff sigma t
Conservative (466 kPa): V = 3.75e-3 466,000 1.5
= 2,621 V per beam
Full pipe (2,847 kPa): V = 3.75e-3 2,847,000 1.5
= 16,014 V per beam

VOLTAGE across relieving chamber stack (5 chambers in series):

Conservative: ~13,000 V (13 kV)
Full pipe: ~80,000 V (80 kV)

CURRENT per beam:

I = d_eff sigma A * omega
Conservative: I = 0.3e-12 466,000 54.8 * 103
= 7.9e-4 A = 0.79mA
Full pipe: = 4.8mA

ELECTRICAL POWER per beam:

Conservative: P = 2,621 * 0.79e-3 = 2.07 W
Full pipe: P = 16,014 * 4.8e-3 = 76.9 W

TOTAL ELECTRICAL (43 beams):

Conservative: ~89 W
Full pipe: ~3,400 W = 3.4kW

SUMMARY TABLE

Parameter | 10 L/s | 100 L/s | 372 L/s

Hydraulic input | 4.6kW | 46.1kW | 171.5kW Acoustic output | 2.8kW | 27.7kW | 102.9kW Power to KC granite| 0.9kW | 8.9kW | 32.9kW KC pressure (raw) | 20.9 kPa| 66.0 kPa| 127.3 kPa KC pressure (Q=500)| 466 kPa | 1,476kPa| 2,847 kPa KC equiv SPL | ~147dB | ~157dB | ~163dB V per granite beam | 2,621 V | 8,295 V | 16,005 V V across stack | ~13 kV | ~41 kV | ~80 kV Piezo current/beam | 0.79mA | 2.5mA | 4.8mA Total elec. power | ~89 W | ~890 W | ~3,400 W

Submitted by Quantitative Analysis — Calculations June 06, 2026

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