LOGOS Learning · 20
Orifice plate calculation step by step (ISO 5167): beta, discharge coefficient and bore
ISO 5167-2 relates mass flow and differential pressure by qm = C / sqrt(1 - beta^4) * (pi/4) * d^2 * sqrt(2 * dP * rho). Because C depends on beta, the bore is found by iteration; for 60 m3/h of water at 25 kPa in a 102.3 mm line it converges to beta = 0.652, C = 0.6097 and d = 66.7 mm.
We want to measure up to 60 m³/h of water in a DN 100 line with an orifice plate and a 0 to 25 kPa transmitter. What bore diameter? The discharge coefficient depends on the very β we are looking for, so the calculation goes in rounds — as ISO 5167 recommends.
- Step 1 of 10
Design data
Full-scale ΔP is a design choice: higher gives more signal, lower gives less permanent loss. 25 kPa (250 mbar) is a common value for water.
Q = 60 m³/h · ΔP = 25 kPa · D = 102.3 mm · ρ = 997 kg/m³ · μ = 0.00089 Pa·s
Formulas in plain text
- Orifice equation (ISO 5167-2)
qm = C / sqrt(1 - beta^4) * (pi / 4) * d^2 * sqrt(2 * dP * rho), beta = d / D- qm = mass flow (kg/s) · C = discharge coefficient (-) · beta = diameter ratio d/D (-) · d = orifice bore (m) · D = pipe internal diameter (m) · dP = differential pressure at full scale (Pa) · rho = density (kg/m3)
- Invariant used to iterate beta
C * beta^2 / sqrt(1 - beta^4) = qm / ((pi / 4) * D^2 * sqrt(2 * dP * rho)) = A ; X = A / C ; beta = (X^2 / (1 + X^2))^(1/4)- A = known side of the equation (-) · start with C = 0.6 and repeat until C stops changing
- Reader-Harris/Gallagher discharge coefficient
C = 0.5961 + 0.0261 * beta^2 - 0.216 * beta^8 + 0.000521 * (1e6 * beta / ReD)^0.7 + (0.0188 + 0.0063 * A_rhg) * beta^3.5 * (1e6 / ReD)^0.3 + tap terms(L1, L2')- ReD = 4 * qm / (pi * D * mu), Reynolds on pipe diameter (-) · A_rhg = (19000 * beta / ReD)^0.8 (-) · L1 = L2' = 25.4 / D(mm) for flange taps (-) · mu = dynamic viscosity (Pa*s)
- Permanent pressure loss
dw = (sqrt(1 - beta^4 * (1 - C^2)) - C * beta^2) / (sqrt(1 - beta^4 * (1 - C^2)) + C * beta^2) * dP- dw = unrecovered pressure loss (Pa)
Frequently asked questions
Why is the orifice plate calculation iterative?
The discharge coefficient C from Reader-Harris/Gallagher depends on beta, and beta depends on C through the orifice equation. Starting from C = 0.6 and alternating the two converges in two or three rounds, as ISO 5167 recommends.
What beta ratio range is allowed by ISO 5167-2?
The equation is valid for 0.1 <= beta <= 0.75, pipe diameters from 50 to 1,000 mm and ReD above 5,000 (above 16,000*beta^2 for beta > 0.56). In practice designers aim for roughly 0.2 to 0.7 to keep good signal and accuracy.
How much pressure does an orifice plate permanently lose?
Only part of the measured dP is recovered downstream; the rest is a permanent loss that adds to the pump head. In the example, with beta = 0.652, it is 14.21 kPa, about 57 % of the 25 kPa differential.