LOGOS Learning · 1
Gravity flow in pipes: water flowing down without a pump
In gravity flow the only energy is the elevation between the reservoir level and the outlet, and the water speeds up until the losses use all of it: dz = (f * L/D + sum K) * v^2 / (2*g). The pipe diameter then decides how much flow passes.
Without a pump, the only energy is the elevation between the reservoir level and the outlet. The pipe decides how much water passes: change the elevation and the diameter.
Quick test
Three questions about this lesson. Got one wrong? The explanation shows right away.
1. Partly closing the valve on a gravity line, the flow…
2. In a gravity line, what sets the flow?
3. Why is a very low velocity in a gravity line also a problem?
Why this happens
Arrows compare with Δz = 10 m at the same diameter.
Energy balance
Water speeds up until the pipe head loss uses up exactly all the available elevation. It is the same crossing as in the pumped lessons, except the "pump curve" became a horizontal line at height Δz.
Flow grows with the square root of elevation
Since loss grows with the square of velocity, doubling the elevation does not double the flow: it rises about 41 %. Doubling the flow would take four times the elevation.
Diameter rules
For the same elevation, a larger diameter passes much more water: flow grows roughly with D to the power 2.5. It is the first design knob on a gravity line.
Continuity
Flow is the cross-section area times the mean velocity. Very high velocities draw the calculator’s attention (erosion, noise, surge); very low ones let solids settle.
Formulas in plain text
- Energy balance of a gravity line
dz = f * (L / D) * v^2 / (2 * g) + sum(K) * v^2 / (2 * g)- dz = elevation from reservoir level to outlet (m) · f = Darcy friction factor (-) · L = pipe length (m) · D = internal diameter (m) · v = mean velocity (m/s) · sum(K) = sum of fitting loss coefficients, entrance and exit included (-) · g = 9.81 m/s2
- Flow versus elevation
Q ~ sqrt(dz)- Q = flow (m3/h)
- Flow versus diameter
Q ~ D^2.5 (same elevation)- D = internal diameter (m)
- Continuity
Q = (pi * D^2 / 4) * v- Q = flow (m3/s) · v = mean velocity (m/s)
Frequently asked questions
How do you calculate gravity flow in a pipe?
Set the available elevation equal to the friction loss plus the fitting losses, dz = (f*L/D + sum K)*v^2/(2g), and solve for the velocity. Because f depends on velocity through the Reynolds number, the solution is iterative; then Q = A*v.
Does doubling the height double the flow?
No. Losses grow with the square of velocity, so doubling the elevation raises the flow by only about 41 % (sqrt(2)). To double the flow you need about four times the elevation.
How much does pipe diameter change gravity flow?
A lot: at the same elevation, flow grows roughly with D^2.5, so a pipe twice as large passes more than five times the flow. Diameter is the first design knob on a gravity line.
Is there a velocity limit in a gravity line?
Yes. Very high velocities bring erosion, noise and water hammer risk, while very low ones let solids settle in the pipe. The diameter is chosen to keep the velocity between those limits.