Underground Cistern Water Engine

What is modelled

The hall is a rectangle with 336 columns standing in the water. For a given depth the model gives the stored volume (hall area minus column sections), the water pressure on the floor and walls, the turning moment on a wall, the upward load on the columns, and how the level changes when water leaves through an outlet and arrives from an aqueduct. The depth, density, hall size and column size are those of the 3D page; wall thickness, brick density, outlet and inflow are added here.

Equations

• Stored volume V = (L W - N π r2) h with L = 138 m, W = 65 m, N = 336, r = 0.425 m.
• Hydrostatic pressure p = ρ g h, thrust per metre ½ ρ g h2, moment about the toe ρ g h3 / 6.
• Wall resisting moment γ H t2 / 2 per metre (brick weight 1900 kg/m3, 9 m high, thickness t); margin = resisting / overturning.
• Outlet flow (Torricelli) Q = Cd Ao √(2 g h), C_d = 0.6; A dh/dt = Qin - Q; emptying time (A / Cd Ao) √(2 h0 / g).

Checks run in the page

• Layer-by-layer volume against the closed form.
• Numerical pressure integral against the thrust and the moment.
• Stepped draining time against the exact Torricelli result.
• Column buoyancy from base pressure against the weight of displaced water.

Honest simplifications

The walls are treated as a free-standing block with no vault, soil or neighbouring walls to help them, so the margin is a teaching number, not a safety assessment of the real monument. The outlet and inflow are invented. Water is incompressible, the level is flat, and leakage, evaporation and silt are ignored.

• Choose Full cistern and then Thin wall, full to see the overturning margin fall below 1.
• Choose Wide outlet draining to watch the level fall faster as the pipe gets bigger.
• Choose Inflow equals outflow: the level settles where the Torricelli outflow matches the inflow.