M095 · Fluid mechanics

Siphon, Pressure Limit & Cavitation

How high can a primed siphon lift the liquid? Only until the absolute pressure at the crest falls to the liquid's vapor pressure. The drop to the outlet drives the flow. Raising the crest adds no drive; it only lowers the pressure up there, and friction lowers it further. At the limit the liquid boils at the top and the column breaks. For water at sea level, the crest can be at most about 10 metres above the surface it draws from.

Raise the siphon crest, change how far the outlet drops, and prime or break the liquid column. Follow the absolute pressure along the tube, see when the vapor-pressure limit rejects the column, and estimate the flow resistance from timed collection.

Inference status $\text{Ready}$

Accepted transport flow $\text{Ready}$

Candidate probe speed $\text{Ready}$

Candidate probe gauge pressure $\text{Ready}$

Candidate probe absolute pressure $\text{Ready}$

Minimum candidate absolute pressure $\text{Ready}$

Minimum margin above vapor pressure $\text{Ready}$

Probe centreline elevation $\text{Ready}$

Tube centreline length $\text{Ready}$

Candidate probe energy head $\text{Ready}$

Candidate probe hydraulic head $\text{Ready}$

Candidate total loss head $\text{Ready}$

Candidate energy balance residual $\text{Ready}$

Acquired stations $\text{Ready}$

Flow inferred from raw readings $\text{Ready}$

Fit coordinate residual $\text{Ready}$

Single-phase status $\text{Ready}$

Single-phase candidate flow $\text{Ready}$

Accepted collected volume $\text{Ready}$

Collection duration $\text{Ready}$

Total resistance inferred from data $\text{Ready}$

Shape the pipe, then measure the pressure

Will this siphon sustain a liquid column?

Crest above source surface

z_c-z_S

2.00\,m

Outlet drop below source

z_S-z_o

1.00\,m

Column filled: no / yes

b

1.00\,

Collection duration

t

10.00\,s

Tube inner diameter

D

40.00\,mm

Reference Darcy friction factor

f_D

0.02\,

Reference total minor loss

K_\Sigma

1.20\,

Volume noise bound

\epsilon_V

0.00\,mL

Probe path fraction

\xi_p

0.50\,

Liquid density

\rho

1000.00\,kg/m^3

Gravity

g

9.81\,m/s^2

Ambient absolute pressure

p_a

101.33\,kPa

Specified vapor pressure

p_v

2.34\,kPa

The source surface stays at a fixed level, and its speed is neglected. The pipe outlet opens to the same atmosphere, and the receiver is a drained sink that keeps a constant level. The collected volume integrates the accepted steady flow over the collection time, and the source never runs down. Tube radii are drawn four times larger than true, but centreline heights are true to scale.

A fixed Darcy factor accounts for wall losses. The total minor-loss coefficient is split equally among the entrance and the two circular bends. The tube starts primed, and the startup flow is not computed. The lowest pressure is checked along the whole tube. If vapor could form, the displayed column empties and acquisition is blocked; bubble growth, dissolved gas and two-phase flow are not modeled.

A straight-line fit of acquired volume against time, with a free intercept, estimates the flow rate and the total resistance. These readings cannot tell wall friction apart from minor losses. Changing the time or probe keeps the records; changing the setup or noise clears them. The CSV leaves out the reference losses. A new reading at the same time replaces the old one, and the noise is bounded and reproducible.

The source level is fixed, the outlet is open to the atmosphere and the losses are prescribed. Priming is set directly, and the vapor pressure serves only as a screening limit. Bubble growth, startup and two-phase flow are not simulated.

Start with the essentials

Focus question: Why does raising the crest put the flow at risk without adding any driving head?
One-sentence intuition: The drop to the outlet supplies the energy, while the crest height and the losses along the way lower the absolute pressure.

Core mathematical model

The outlet drop sets the speed

\begin{aligned}v&=\sqrt{\frac{2g(z_S-z_o)}{1+f_D L/D+K_\Sigma}}\\Q&=Av\end{aligned}

The term for the kinetic energy leaving the outlet stays in the formula, even in the no-loss comparison.

Absolute pressure along the tube

\begin{aligned}p_{\mathrm{abs}}(s)&=p_a+\rho g\left[z_S-z(s)\right]\\&\quad-\frac{\rho v^2}{2}-\rho g h_L(s)\end{aligned}

With friction and bend losses, the lowest pressure can sit just after the crest.

Three conditions for flow

\begin{gathered}\text{primed},\quad z_S>z_o,\\\min_s p_{\mathrm{abs}}(s)>p_v\end{gathered}

All three must hold. A rejected solution is still drawn as a diagnostic curve, but no volume is collected.

Flow from timed collection

V_i=b+\widehat Q t_i

A free intercept absorbs any sensor tare offset, and only the acquired readings enter the fit.

Only the total resistance can be found

\widehat R=\frac{2g(z_S-z_o)}{(\widehat Q/A)^2}-1

Collected volumes cannot separate Darcy friction from minor losses.

Common difficulties

Gauge and absolute pressure

Typical misconceptionA negative gauge pressure always means the liquid cannot flow.

Better mental modelAdd the ambient pressure first, then compare the result with the specified vapor pressure. The check covers the whole path.

The fitted resistance is an estimate

Typical misconceptionThe fitted coefficient is just the reference slider under another name.

Better mental modelClear the record and the estimate disappears. Noise can even give an unphysical negative coefficient: the fit is unconstrained, so the result is not clipped, and it is not a safety certificate.

Run the experiment

01
Prime the column
Choose Unprimed tube, then press Prime the tube.
What to observe: An unprimed tube carries no flow, even when the available energy would allow it.
02
Move the outlet
Lower the outlet and compare with No-loss comparison.
What to observe: The outlet drop sets the speed. Raising the crest alone does not drive the ideal flow.
03
Find the pressure limit
Raise the crest or lower the ambient pressure, and look at the marked station with the lowest pressure.
What to observe: Once the pressure there reaches the specified vapor pressure, the column is rejected.
04
Collect liquid and estimate
Return to Primed siphon, press Acquire timed volumes, then press Export raw readings.
What to observe: The slope of volume against time gives the flow and the total resistance, but it cannot separate the different loss mechanisms.