Physics calculator

ISO 5167 Long-Radius Nozzle Calculator

Compute ISO 5167-3 long-radius nozzle discharge with C from β. Runs locally. Not ISA 1932, not given C_d=0.98, not RHG orifice, not ISO 5167-4 type C, not Q=Av.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 5/5 tests · Production surface contract 1/1 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: ISO 5167-3 long-radius nozzle discharge from D1, D2 or β, Re_D, and Δp. Invert Δp. Report C from β.
Scope: ISO 5167-3 long-radius nozzle: Q = C E A2 √(2Δp/ρ); C = 0.9965 − 0.00653 β^{0.5}; incompressible ε=1; C is reported, not an input
Verification: Engine tested · 5/5 tests · Production surface contract 1/1 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 80000-4 — Mechanics — Length, speed, pressure
ISO 5167-3 — Measurement of fluid flow by means of pressure differential devices — Nozzles and Venturi nozzles — Long-radius nozzle discharge coefficient
Sources
Evidence: 5 property · Production surface contract 1/1 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-18T02:44:05.986Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

DischargeQ = C E A2 √(2Δp / ρ)

Approachβ = D2/D1 · E = 1/√(1 − β⁴)

Long-radius CC = 0.9965 − 0.00653 β^{0.5}

iSI. Incompressible ε=1. C is reported from β; do not send C_d. Do not send query v — that is calculation_version. Re_D bounds the envelope; it does not interpolate C.

How to use

Enter D1 and D2

Inlet D1 and throat D2. REST may send D1+beta instead of D2. This is not an orifice d+D pair.

Enter Re_D

Pipe Reynolds number on D1. Required as the ISO envelope. Do not send ρ, μ, and v to wrap Reynolds.

Enter Δp and ρ

Inlet-to-throat differential pressure. Invert on the Δp tab. Read C on the C tab without Δp.

Example calculations

Common configurations with formula and result.

β=0.5, Re_D=10⁶, 10 kPa

D1=0.1 m, D2=0.05 m, Δp=10 kPa, ρ=1000 kg/m³

C=0.9965−0.00653√β, Q=C E A2 √(2Δp/ρ)

C≈0.99188 · Q≈0.008995 m³/s

Same β via D1+beta

D1=0.1 m, beta=0.5

D2=β D1

same C

ISO 5167 Long-Radius Nozzle calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 5/5 tests · Production surface contract 1/1
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: ISO 5167-3 long-radius nozzle. Incompressible Q = C E A2 √(2Δp/ρ) with E=1/√(1−β⁴). C = 0.9965 − 0.00653 β^{0.5}. Re_D is the ISO envelope, not a term in C. β∈[0.20,0.80], D∈[50 mm, 630 mm], Re_D∈[1e4, 1e7]. Not ISA 1932 C(β,Re), not a given C_d=0.98, not an RHG orifice plate, not ISO 5167-4 type C.
What it calculates: ISO 5167-3 long-radius nozzle discharge from D1, D2 or β, Re_D, and Δp. Invert Δp. Report C from β.
Inputs: mode?
D1
D2|beta
Re_D
dP?
rho?
Q?
Outputs: Q?
D1
D2
beta
E
A1
A2
C
Cd
Re_D
v1?
v2?
rho?
dP?
mode
model
Formula: Q = C E A2 √(2Δp/ρ); C = 0.9965 − 0.00653 β^{0.5}
Assumptions: ISO 5167-3 long-radius nozzle: Q = C E A2 √(2Δp/ρ); C = 0.9965 − 0.00653 β^{0.5}; incompressible ε=1; C is reported, not an input
C from β, not Re_D; Re_D∈[1e4,1e7] is the ISO envelope; β∈[0.20,0.80]; D∈[50 mm, 630 mm]
Not ISA 1932 C(β,Re), not given C_d=0.98, not RHG orifice, not ISO 5167-4 type C, not Q=Av, not Darcy f, not compressible ε
Units: SI; D1, D2 in m; Q in m³/s; Δp in Pa; ρ in kg/m³; Re_D dimensionless
Boundary conditions: missing D1, D2/beta, Re_D, Δp, Q, or ρ when required → MISSING_REQUIRED_INPUT
D outside [50 mm, 630 mm], β outside [0.20,0.80], or Re_D outside [1e4,1e7] → VALUE_OUT_OF_RANGE
C_d given, orifice d+D, tank H, A or v, Darcy f, μ, Cv/Kv, K, ε, tapping, kind=isa1932 or kind=machined → MIXED_INPUT_ENCODING
mode=Cd or unknown mode → INVALID_MODE
Example: D1=0.1 D2=0.05 Re_D=1e6 dP=10000 rho=1000 → C≈0.99188 Q≈0.008995
Validation cases: 4 published on this page · 5/5 tests · Production surface contract 1/1 · View evidence
D1=0.1 D2=0.05 Re_D=1e6 dP=10000 rho=1000 → C≈0.99188 Q≈0.008995
D1=0.1 D2=0.05 Re_D=1e6 Cd=0.98 → MIXED_INPUT_ENCODING
D1=0.1 d=0.05 D=0.1 Re_D=1e6 dP=10000 rho=1000 → MIXED_INPUT_ENCODING
mode=Cd D1=0.1 D2=0.05 Re_D=1e6 → INVALID_MODE
Sources: ISO 80000-4 — Mechanics — Length, speed, pressure
Supports: D is a length; Q is a volume flow; Δp is a pressure
ISO 5167-3 — Measurement of fluid flow by means of pressure differential devices — Nozzles and Venturi nozzles — Long-radius nozzle discharge coefficient
Supports: C = 0.9965 − 0.00653 β^{0.5}. Not ISA 1932, not a given C_d, not an orifice plate, not ISO 5167-4 type C
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute ISO 5167-3 long-radius nozzle discharge from β, Re_D, and Δp. C is from β, not supplied.

Supported and not supported

Supported — Q from D1, D2 or β, Re_D, and Δp · invert Δp · C from β · API physics.fluid.long_radius_nozzle

Not supported — ISA 1932 C(β,Re), given C_d=0.98, RHG orifice, ISO 5167-4 type C, Q=Av, Darcy f, compressible ε

Agent / API notes

Capability id: physics.fluid.long_radius_nozzle · tool id: long-radius-nozzle · pin 1.0.0.

{ "D1": 0.1, "D2": 0.05, "Re_D": 1000000, "dP": 10000, "rho": 1000 }

Modes: flow (default), pressure, coefficient. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/long-radius-nozzle. There is no /calc/fluid.

Other calculators in this family: ISO 5167 nozzle, ISO 5167 Venturi, Venturi meter, ISO 5167 orifice .

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Frequently asked questions

Key distinctions behind the calculation.

Is this an ISA 1932 nozzle?

No. ISO 5167-3 ISA 1932 is /calc/physics/iso-nozzle (C from β and Re_D, ≈0.97656 at β=0.5 and Re_D=10⁶). Sending kind=isa1932 here is MIXED_INPUT_ENCODING. This page reports C≈0.99188 at the same β.

Is this the given-C_d venturi page?

No. A classical venturi with a given C_d (default 0.98) is /calc/physics/venturi. Sending C_d here is MIXED_INPUT_ENCODING.

Is this an ISO 5167-4 Venturi tube?

No. Machined/rough/as-cast type C is /calc/physics/iso-venturi (machined C=0.995). Sending kind=machined here is MIXED_INPUT_ENCODING.

Is this an ISO 5167-2 orifice plate?

No. Reader–Harris/Gallagher C from β, Re_D, and tapping is /calc/physics/iso-orifice. Sending d and D is MIXED_INPUT_ENCODING.

Does C change with Re_D?

No. Long-radius C is a function of β only. Re_D must lie in [1e4, 1e7] or the engine returns VALUE_OUT_OF_RANGE. ISA 1932 C does depend on Re_D; that is /calc/physics/iso-nozzle.

Is this Q = A·v?

No. Volumetric flow from a given area and velocity is /calc/physics/flow-rate. Sending A or v is MIXED_INPUT_ENCODING.

Where does this run?

Locally in the browser by default. REST and MCP call the same fluid-engine longRadiusNozzle wrapper.