LOGOS Learning · 18
Gravity flow calculation step by step: flow rate, head loss and pipe size
With no pump, the elevation is all spent on losses: dz = (f * L/D + sumK) * v^2/(2g), so v = sqrt(2*g*dz / (f*L/D + sumK)). Since f depends on Reynolds and Reynolds on v, the calculation iterates; for 10 m of fall through 200 m of DN 100 it converges in four rounds to 66.4 m3/h.
A reservoir whose level is 10 m above the outlet feeds, by gravity, a 200 m DN 100 line. How much water flows? With no pump, the flow is the one whose head loss uses up exactly the available elevation — and since friction depends on the flow itself, the calculation needs a few rounds.
- Step 1 of 9
Energy balance
Between the reservoir level and the free outlet, the only energy is the elevation. It turns into major and minor losses; the exit velocity head is the K of the free exit (K = 1).
Δz = f · LD · v22g + ΣK · v22gΔz = 10.0 m · L = 200 m · D = 0.1023 m
Formulas in plain text
- Energy balance (reservoir to free outlet)
dz = (f * L / D + sumK) * v^2 / (2 * g)- dz = elevation from reservoir level to outlet (m) · f = Darcy friction factor (-) · L = pipe length (m) · D = internal diameter (m) · sumK = sum of fitting coefficients, free exit K = 1 included (-) · v = velocity (m/s) · g = 9.81 m/s2
- Velocity solved from the balance
v = sqrt( 2 * g * dz / (f * L / D + sumK) )- v = pipe velocity for the current f (m/s)
- Swamee-Jain friction factor (iterated)
f = 0.25 / [ log10( eps / (3.7 * D) + 5.74 / Re^0.9 ) ]^2, Re = rho * v * D / mu- eps = absolute roughness (m) · Re = Reynolds number (-) · rho = density (kg/m3) · mu = dynamic viscosity (Pa*s) · start with f0 = 0.02 and repeat until f stops changing
- Flow rate (continuity)
Q = (pi * D^2 / 4) * v- Q = flow rate (m3/s; x 3600 for m3/h)
Frequently asked questions
Why does gravity flow need iteration?
The friction factor depends on the Reynolds number, which depends on the velocity you are trying to find. Starting from f = 0.02 and recomputing v, Re and f a few times converges quickly, usually in three or four rounds.
How do I choose the pipe size for a gravity line?
Repeat the calculation for each commercial diameter with the same elevation and pick the smallest one whose flow meets the target. In the example a 40 m3/h target rules out DN 80 (32.7 m3/h) and is met by DN 100 (66.4 m3/h).
Does the exit velocity head count as a loss?
Yes. At a free outlet the water leaves with kinetic energy v^2/(2g) that is not recovered, which is why the free exit enters the balance with K = 1.