Self-cleansing velocity: the formula, and what 1 m/s hides

The formula, first

The velocity at which a deposited particle starts to move is not a constant. It has an equation — Camp's equation, derived from the Shields criterion, given by Metcalf & Eddy as equation 9.1:

V_H = [ 8 · k · (s − 1) · g · d / f ]^1/2

Every term in it is something you either measure or already have. Nothing new enters the problem.

Note f: it is the same friction factor you already compute for head loss. If you have the head loss, you have the self-cleansing velocity.

What the equation actually gives

These are not quoted numbers. They are the equation evaluated at f = 0.025, which sits in the middle of the usual range:

All values in m/s.

Same particle size (1 mm) in every bar. The only thing that changes is what the particle is made of — and the rule of thumb sits far above all four.

Where the «minimum 1 m/s» rule comes apart

Almost everyone sizes a line with one number in mind: keep it above one metre per second. It works most of the time, which is exactly why it survives. But it is not the result of any calculation, and the table above shows which way it fails.

It fails by asking for too much. One-millimetre sand starts moving at 0.46 m/s. A biological floc starts moving at 0.10 m/s. For 1 m/s to be the real threshold you would have to be dragging particles of about 4.8 mm — gravel, not sand. With biological floc, about 106 mm, which is not a particle at all.

So the rule of thumb rarely lets solids settle. What it does, quietly and every hour of every day, is charge you pumping energy for a margin nobody calculated. On a line running continuously, the difference between 1.0 and 0.6 m/s is not a rounding decision — head loss scales with the square of velocity.

The variable that decides everything is (s − 1)

This is what the round number hides. A mineral particle at s = 2.65 gives (s − 1) = 1.65. An organic floc at s = 1.05 gives 0.05. That is thirty-three times less, and under the square root it becomes roughly a factor of six in the required velocity.

Which is why a line carrying biological sludge and a line carrying grit should never be sized against the same threshold — and yet, with the rule of thumb, they almost always are.

What this article does not claim

Two limits, stated plainly, because the difference matters in practice:

• This is incipient motion, not deposition. The equation gives the velocity at which a particle already sitting on the invert begins to move. The velocity below which a particle in suspension settles out is a different question with a different answer.
• Modern gravity-sewer practice uses tractive force, not velocity. What lifts a particle off the invert is the shear stress the water exerts on it, not the mean velocity of the flow — velocity is a proxy that happens to work for common geometries. We are not quoting shear-stress design limits here because we do not have an open, contrastable source for them in front of us. When we do, we will cite it.

Using it on a real line

1. Measure d. It is the only figure you cannot look up. A sieve analysis of what the line actually carries beats any table.
2. Take f from your head-loss calculation, not from a default. It moves the answer by about 10 % across its range.
3. Compute V_H, then add your own margin — and know that you are adding it. That is the whole point: a margin you chose is engineering, a margin you inherited is folklore.

Our friction-loss calculator gives you f for the pipe and flow you actually have, which is the input this equation needs.

Source

Camp's equation as given by Metcalf & Eddy, Wastewater Engineering, equation 9.1, derived from the Shields criterion for incipient particle motion. The constants k, s and f are those stated by that source. The values in the table were computed from the equation itself, not transcribed.