Flow delivered
– m³/h
Total ΔP dissipated
– bar
Min cavitation index σ
–
Predicted noise
– dB(A)
Side view — horizontal orifice train (flow left → right)
No cavitation (σ ≥ 1.5)
Incipient (1.0–1.5)
Moderate (0.8–1.0)
Severe (< 0.8)
Distance between orifices (× D)— linked live with the “Orifice spacing (live)” tab
Staged pressure profile through the train
Vena contracta — local pressure & cavitation across the whole train
Local pressure across the whole arrangement: it dips sharply at each orifice (the vena contracta) and recovers in the run to the next orifice. Recovery is gradual — the static pressure climbs back over ~several pipe diameters (φ = 1 − e^(−L/2D): ~95% by 6×D). If the orifices are spaced too closely the flow has not re-pressurised before the next restriction, so the local upstream pressure — and the cavitation margin σ — is reduced and the dips deepen toward vapour pressure (red = flashing). Bring the spacing slider down and the downstream orifices cavitate; open it out and they recover. Recovery F_L also rises with a 45° chamfer and orifice thickness, both of which reduce cavitation. P_vc = P_up,local − ΔP / F_L².
Orifice geometry — plan (face) & section (flow) view
Per-orifice results
Methods & references — science & math used at each step
Every calculation in this tool, the equation as implemented, and its source. Primary standards are ISO 5167 (orifice metering), IEC 60534 (control-valve sizing, recovery & noise) and ISA cavitation practice.
1 · Discharge coefficient & pressure drop
| Step / quantity | Equation as implemented | Scientific basis / reference |
|---|---|---|
| Discharge coeff Cᵈ (sharp) | Cᵈ = 0.5961 + 0.0261β² − 0.216β⁸ + 0.000521(10⁶β/Re)^0.7 + (0.0188+0.0063A)β^3.5(10⁶/Re)^0.3 + small-bore term | Reader-Harris–Gallagher equation; ISO 5167-2:2003; Reader-Harris, Orifice Plates & Venturi Tubes, Springer 2015 |
| Cᵈ (45° conical entrance) | Cᵈ ≈ 0.73–0.75 (streamlined inlet) | ISO 5167-4 (conical-entrance orifice) |
| Orifice ΔP (incompressible) | ΔP = (ρ/2)(1−β⁴)(Q/(Cᵈ·A₀))², β=d/D | ISO 5167-1/-2; Miller, Internal Flow Systems, 2nd ed., BHRA 1990; Crane TP-410 |
| Reynolds number | Re = ρ·v·D/μ | Fluid mechanics (pipe-bore basis) |
| Series flow (self-limiting) | Q_op = Q·√(ΔP_avail/ΣΔP_design) | Series hydraulic resistances; energy balance |
2 · Cavitation & pressure recovery
| Cavitation index σ | σ = (P_down − Pᵥ)/ΔP (must stay ≥ 1.5; ≥ 1.0 multi-hole) | ISA RP75.23-1995; Tullis, Hydraulics of Pipelines, Wiley 1989; NORSOK L-001 |
| Recovery factor F_L | P_vc = P_up − ΔP/F_L² (vena-contracta pressure) | IEC 60534-2-1; ISA S75.01 (liquid pressure-recovery factor) |
| Interstage recovery (this tool) | φ = 1 − e^(−L/2D); deficitᵢ = (1−φ)[deficitᵢ₋₁ + ΔPᵢ₋₁(1/F_L²−1)] | Screening model from orifice recovery-length data (~6–8 D): Miller 1990; ESDU 81039; tool derivation |
| Equal-σ staging | ΔPᵢ = (P_up,i − Pᵥ)/(σ_design + 1) | Derived from the σ definition (equal-index split) |
| Bore from ΔP | v=√(2ΔP/ρ) → A₀=Qₛ/(Cᵈv) → d=√(4A₀/π) | ISO 5167-2 orifice equation, inverted |
| Vapour pressure (seawater) | Pᵥ ≈ 0.0234·e^(0.0612(T−20))(1−0.00066·S) | Seawater correlation; Sharqawy, Lienhard & Zubair, Desal. & Water Treat. 16:354, 2010 |
3 · Multi-hole geometry & noise
| Multi-hole equivalence | d_eq = √N · d_hole (total open area sets ΔP) | Total-area basis; IEC 60534 multi-path trim |
| Hydrodynamic noise (liquid) | U_vc=√(2ΔP/ρ)/F_L · W_m=½ṁU_vc² · η∝(ΔP/P_up)^0.8·β^−0.5·(0.63/F_L)²·f_cav · L_W=10log₁₀(ηW_m/10⁻¹²) → SPL at 1 m | BS EN 60534-8-4 (liquid noise); BS EN 60534-8-3 (pipe-wall TL) |
| Transmission loss | TL = 20·log₁₀(t) − 10·log₁₀(OD/100) + C_mat + 3·log₂(N) | BS EN 60534-8-3 pipe-wall TL; NORSOK S-002 (limits) |
| Multi-hole noise credit | ΔTL = 3·log₂(N) (smaller jets) | Jet-size scaling; IEC 60534-8-3 |
4 · Plate geometry
| Square edge + downstream bevel | sharp upstream edge; 45° bevel if E > e (thick plate) | ISO 5167-2 §5.1.5 (plate thickness & bevel) |
| Conical entrance (chamfer) | 45° converging inlet cone → cylindrical throat (raises Cᵈ & F_L) | ISO 5167-4 |
| Rounded (radiused) bore | quarter-circle inlet, radius r: as r/d rises the vena contracta is suppressed → quarter-circle Cᵈ≈0.78 (r/d≈0.2), → ISA-1932 / long-radius nozzle Cᵈ≈0.95–0.97, F_L→0.93. d+2r ≤ D; rᵤ+r_d ≤ t. | ISO/TR 15377, BS 1042-1.2 (quarter-circle orifice); ISO 5167-3 (nozzles) |
| Assembly length | L = (N−1)·spacing·D + N·t | Geometric build-up; ISO 5167 straight-run requirements for end runs |
Screening tool — the cavitation σ thresholds, the interstage-recovery model and the simplified IEC noise method are indicative; verify against the full standards (ISO 5167, IEC 60534-8-3/8-4) and vendor data for critical service.
Orifice spacing — set every gap, watch the physics live
Choose the number of orifices and the distance to the next orifice for each individual gap (drag the slider or type, in × pipe-diameters D — the mm value is shown). Every change feeds straight into the cavitation physics: the local upstream pressure, ΔP, vena-contracta pressure P_vc and σ for every orifice update in real time below. The orifice count and these gaps are stored with the rest of the setup and used by all the other tabs.
Per-gap interstage recovery: φᵢ = 1 − e^(−Lᵢ/2D) for each individual gap Lᵢ; the un-recovered head accumulates as a cumulative deficit down the train, so tightening any one gap depresses every downstream orifice's σ. Two thresholds — don't confuse them: the verdict/status uses the design margin σ ≥ 1.5 ("Incipient" below it = cavitation can begin), while the red line is the hard physical limit P_vc = Pᵥ (full flashing). A dot can read "Incipient" yet still sit above the red line — bubbles/noise have started but the minimum pressure hasn't reached vapour yet. Note: if σ_min stays below 1.5 even at wide spacing, the limit is the stage count (too few orifices for the pressure ratio), not the gap — opening the spacing won't help; add orifices or raise the downstream pressure. Screening model — confirm against CFD/test for critical service.
Cavitation proof — working (upstream) vs downstream, step by step
The minimum pressure in the flow is the vena-contracta pressure just downstream of each orifice. Cavitation begins the instant it reaches the vapour pressure Pᵥ. Criterion: no cavitation ⟺ P_vc > Pᵥ at every orifice, where P_vc = P_up,local − ΔP / F_L². The only thing that differs below is the pressure-recovery factor F_L (upstream cone ≈ 0.81, downstream bevel ≈ 0.68). Updates live with N, spacing and conditions.
Same orifice train, same equal-σ ΔP split — only the inlet geometry (F_L) changes. The streamlined upstream cone keeps every vena-contracta minimum above Pᵥ; the downstream chamfer lets the last stages fall below it. Screening tool — verify against ISO 5167 / IEC 60534 for critical service.
Stress analysis — orifice plate as a clamped diaphragm under ΔP
Each orifice plate is a flat circular plate clamped at the pipe bore (between the flanges) with its central bore open, carrying the differential pressure ΔP across its faces. The static line pressure acts on both faces and cancels — the net structural load is ΔP. Stresses are computed per orifice with Roark/Timoshenko flat-plate theory (the inlet plate, with the largest ΔP, governs).
Stress field: flat circular plate, outer edge clamped at the pipe bore (span = pipe ID), uniform differential pressure, small-deflection (Kirchhoff) plate theory — Roark's Formulas for Stress and Strain Table 11.2 case 10a & Timoshenko Theory of Plates and Shells; central bore treated as a stress raiser (Goodier bending SCF). Code check to British/European standards: the required thickness is verified to PD 5500 (BS 5500) §3.5.5 and BS EN 13445-3 Clause 10 (flat plate, e = C·D·√(P/f)) using the BS/EN nominal design stress f = min(Rp0.2/1.5, Rm/2.4). Screening calculation — for full compliance add the corrosion allowance, the bolted/gasketed edge-moment term, and a finite-element check of the actual perforation and clamp; cross-check ASME VIII Div. 2 Part 5 if working to ASME.