Siphon (3D)

Why a siphon works

Once the hose is full, the liquid is a continuous chain. The downhill leg is longer than the uphill leg, so gravity pulls the longer column down harder and the chain slides over the crest. Air pressure does not push the liquid over the top; it keeps the chain intact by pushing water up the intake against the pressure that falls at the crest.

Bernoulli with losses

From the upper surface (at air pressure, speed about zero) to the outlet, the head available is dH = h1 - h_out. Energy balance: g dH = (v^2 / 2) (1 + K_in + K_bends + f L / D). The 1 is the kinetic energy that leaves with the water, K_in = 0.5 is the entrance loss and each sharp bend adds 0.4. The friction factor is f = 64 / Re for laminar flow and f = 0.316 Re^(-1/4) (Blasius, smooth pipe) for turbulent flow, whichever is larger. The speed is found by bisection because f depends on v.

Pressure at the crest

Applying Bernoulli from the surface to the crest at height h_c above the water: p_c = p_air - rho g h_c - (rho v^2 / 2) (1 + K_in + f L_up / D). Raising the crest or speeding up the flow lowers p_c.

The atmospheric limit and cavitation

The liquid cannot pull on itself without limit: when p_c falls to the vapour pressure p_v the liquid boils (cavitates) and gas fills the crest. With no flow the highest crest is h_max = (p_air - p_v) / (rho g): 10.1 m for cold water at sea level, 8.2 m for glycerine (denser, but almost no vapour pressure), 5.7 m for 80 C water, and only 0.76 m for mercury. In the model, if the head-driven speed would push p_c below p_v, the flow is limited (choked) to the speed at which p_c = p_v; above h_max the siphon breaks and must be primed again.

Levelling out

With tank areas A1, A2 the head obeys d(dH)/dt = -Q (1/A1 + 1/A2). For a constant loss coefficient K, Q = a sqrt(2 g dH / K), so sqrt(dH) falls linearly in time and the tanks level out in t = 2 sqrt(dH0) / (a sqrt(2g/K) (1/A1 + 1/A2)). The siphon stops when the levels meet or when the intake is uncovered and sucks air.

Simplifications

Smooth pipe friction, steady (quasi-static) flow at each instant, no dissolved-gas release before cavitation, no hose elasticity, fixed intake and outlet heights in the tanks, and the water jet to the floor is only drawn schematically. Time lapse speeds up the tank filling but not the particle motion.

1. Run the default preset and watch the first graph: the two levels approach each other and the flow stops when they meet.
2. Open the pressure graph: it dips at the crest. Raise the crest with the slider and watch the minimum fall toward the red vapour-pressure line.
3. Use Crest near the 10 m limit, then Crest too high: the siphon refuses to run, or breaks, above about 10 m.
4. Pick Hot water boils at the crest: the flow is choked and white vapour bubbles appear at the top.
5. Try Mercury: with a density 13.5 times that of water the limit drops to 0.76 m.
6. Choose Thin air at 5000 m and compare the pressure minimum with sea level.
7. Press Break it, then Prime the siphon, to see the water legs settle at the tank levels.