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Concentric Reducer Calculator

Compute concentric-reducer loss h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁) referenced to the downstream speed. Runs locally. Not sudden expansion, not gradual contraction α, not vena-contracta C_c, not catalog K, and not Q=Av.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁); plus pressure and velocity solves; + O3 mpmath tabulated head.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁); plus pressure and velocity solves; + O3 mpmath tabulated head.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (head, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not C_c, not α, not (v1−v2)² sudden contraction, not catalog K, not gradual contraction, and not Q=Av.
Numerical scope
O2: concentric reducer K=0.5(1−A2/A1) vs a separate-module identity (≤2 ULP). Not C_c, not α, not (v1−v2)², not catalog K, not gradual contraction, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated concentric-reducer vectors. Not catalog K, not pipe-entrance geometry, not sudden expansion (v1−v2)², not Darcy, and not Q=Av.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Concentric reducer head from downstream speed and area ratio A₂/A₁. Invert v₂. Optional Δp from ρ.
Scope
Concentric reducer: h = K v₂² / (2g); K = 0.5 (1 − A₂/A₁); K referenced to downstream speed; K is reported, not an input
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁); plus pressure and velocity solves; + O3 mpmath tabulated head.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision
2026-09-26.o2-o3 · 1/1 property · digest 7983105f1554
Legacy regression
6/6 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 1/1 CVP boundary · 2/2 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 5/5 CVP contract · Manifest
Sources
  • ISO 80000-4 — Mechanics — Length, speed, pressure
  • Idelchik — Handbook of Hydraulic Resistance — Concentric reducer; K referenced to the downstream section
Sources
Evidence
1 legacy golden · legacy regression suite · 4/4 oracle-backed golden · 2/2 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Head lossh = K v₂² / (2g)
Area-ratio KK = 0.5 (1 − A₂/A₁)
Pressure dropΔp = ρ K v₂² / 2
iSI. v_m_s is the downstream speed v₂. area_ratio is A₂/A₁, not cavitation σ and not venturi β. g defaults to 9.80665 m/s². Do not send query v — that is calculation_version.

How to use

1

Enter the area ratio

area_ratio = A₂/A₁ in (0, 1). REST may send A1+A2 or D1+D2 instead. Do not mix encodings.

2

Enter v₂

v_m_s is the downstream speed. Invert on the v tab from a known h.

Example calculations

Common configurations with formula and result.

ϟ

σ=0.5, v₂=2 m/s

Downstream area is half the upstream area

K=0.5(1−0.5)=0.25, h=K v₂²/(2g)
h≈0.05099 m · K=0.25 · Δp=500 Pa at ρ=1000
ϟ

Same σ via A1+A2

A1=2 m², A2=1 m²

σ=A2/A1=0.5
K=0.25

Concentric Reducer calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Concentric reducer. K is referenced to the downstream speed v₂ and reported from the area ratio σ = A₂/A₁. K = 0.5 (1 − σ). σ ∈ (0, 1). Geometry from area_ratio or A1+A2 or D1+D2. Not vena-contracta C_c, not conical α, not Borda–Carnot (v1−v2)², not a catalog fitting K.
What it calculates
Concentric reducer head from downstream speed and area ratio A₂/A₁. Invert v₂. Optional Δp from ρ.
Inputs
  • mode?
  • area_ratio|A1+A2|D1+D2
  • v_m_s|h
  • rho?
  • g?
Outputs
  • h
  • v
  • v2
  • K
  • area_ratio
  • g
  • rho
  • dP
  • mode
  • model
Formula
h = K v₂² / (2g); K = 0.5 (1 − A₂/A₁)
Assumptions
  • Concentric reducer: h = K v₂² / (2g); K = 0.5 (1 − A₂/A₁); K referenced to downstream speed; K is reported, not an input
  • A₂/A₁ in (0, 1); default g=9.80665 m/s²; default ρ=1000 kg/m³; incompressible; geometry from area_ratio or A1+A2 or D1+D2
  • Not sudden expansion (v1−v2)², not gradual contraction α, not vena-contracta C_c, not catalog minor-loss K given, not a four-way cross, not Q=Av
Units
  • SI; v₂ in m/s; h in m; area_ratio dimensionless; ρ in kg/m³; Δp in Pa; K dimensionless
Boundary conditions
  • missing v_m_s, h, or area_ratio when required → MISSING_REQUIRED_INPUT
  • v_m_s ≤ 0, h ≤ 0, or area_ratio ≤ 0 → VALUE_MUST_BE_POSITIVE
  • area_ratio ≥ 1 or K=0 on velocity invert → VALUE_OUT_OF_RANGE
  • mixed area_ratio with A1+A2 or D1+D2, C_c, α, v1, K given, Q, f, kind, sigma, beta → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
v_m_s=2 area_ratio=0.5 → h=K v₂²/(2g) K=0.25
Validation cases

3 published on this page · 6/6 tests · Production surface contract 5/5 · View evidence

  • v_m_s=2 area_ratio=0.5 → h=K v₂²/(2g) K=0.25
  • v_m_s=2 A1=2 A2=1 → K=0.25
  • v_m_s=2 area_ratio=0.5 C_c=0.62 → MIXED_INPUT_ENCODING
Sources
  • ISO 80000-4 — Mechanics — Length, speed, pressure
    Supports: h is a head; v is a speed; Δp is a pressure
  • Idelchik — Handbook of Hydraulic Resistance — Concentric reducer; K referenced to the downstream section
    Supports: K=0.5(1−A₂/A₁) on v₂. Not vena-contracta C_c, not conical α, not catalog K
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute concentric-reducer loss from downstream speed and the area ratio A₂/A₁.

Supported and not supported

Supported — h from v₂ and A₂/A₁ · invert v₂ · API physics.fluid.concentric_reducer

Not supported — vena-contracta C_c, conical α, Borda–Carnot (v1−v2)², catalog minor-loss K, four-way cross, Darcy f, Q=Av, cavitation σ, venturi β

Agent / API notes

Capability id: physics.fluid.concentric_reducer · tool id: concentric-reducer · pin 1.0.0.

{ "v_m_s": 2, "area_ratio": 0.5 }

Modes: head (default), pressure, velocity. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/concentric-reducer. There is no /calc/fluid.

Other calculators in this family: Alternate Depth Calculator, Bernoulli Equation Calculator, Broad-Crested Weir Calculator, Cavitation Number Calculator, Chezy Formula Calculator, Colebrook Friction Factor Calculator, Critical Depth Calculator, Critical Slope Calculator . Explore all Fluid Mechanics.

→ Alternate Depth Calculator Compute both rectangular depths at the same specific energy from E, Q, and width b. Runs locally. Not specific energy from y+v; not specific force M; not hydraulic jump y2; not critical depth yc as the product. › → Bernoulli Equation Calculator Incompressible Bernoulli along a streamline: solve P2, v2, or h2. Runs locally in your browser. › → Broad-Crested Weir Calculator Compute rectangular broad-crested weir discharge Q = C_d (2/3)^{3/2} b √g H^{3/2}. Default C_d=0.85. Runs locally. Not a thin-plate weir, not Manning, not Q=Av, and not an orifice. › → Cavitation Number Calculator Compute the cavitation number σ = (p − p_v) / (½ ρ v²). Runs locally. Not NPSH, not Thoma NPSH/H, not Weber, and not Euler Eu=Δp/(ρ v²). › → Chezy Formula Calculator Compute open-channel mean velocity V = C √(R S). Runs locally. Not Manning n, not Q=Av, not Hazen–Williams, and not Darcy. › → Colebrook Friction Factor Calculator Solve the Colebrook–White equation for Darcy friction factor f from Re and ε/D. Turbulent pipes Re≥4000. Runs locally. Not laminar 64/Re and not a Moody-chart lookup. › → Critical Depth Calculator Compute rectangular critical depth yc = (q²/g)^{1/3}. Runs locally. Not a Froude wrap, not specific energy, not a hydraulic jump, and not Q=Av. › → Critical Slope Calculator Compute rectangular critical bed slope Sc such that yn=yc using SI Manning at y=yc. Runs locally. Not Manning A,R; not normal-depth yn; not critical-depth yc as the product; not Chezy; not Q=Av. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a sudden contraction?

No. Vena-contracta sudden contraction is /calc/physics/sudden-contraction. That page takes C_c (default 0.62) and does not take A₂/A₁. Sending C_c here is MIXED_INPUT_ENCODING. At σ=0.5 this page reports K=0.25; sudden contraction still reports K≈0.376.

Is this a gradual contraction?

No. The conical confuser with wall half-angle α is /calc/physics/gradual-contraction. Sending alpha_deg is MIXED_INPUT_ENCODING.

Is this sudden expansion?

No. Borda–Carnot (v1−v2)² is /calc/physics/sudden-expansion. This page is a reducer, A₂<A₁, and uses one downstream speed.

Is this minor-loss K?

No. Catalog fitting K is /calc/physics/minor-loss. Sending K is MIXED_INPUT_ENCODING. This page reports K from A₂/A₁.

Is this Q = A·v?

No. Volumetric flow is /calc/physics/flow-rate. Sending Q or a single A is MIXED_INPUT_ENCODING.

Where does this run?

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