Physics calculator

Gradual Expansion Calculator

Q: Is this minor-loss K?

No. Catalog fitting K is /calc/physics/minor-loss. Sending K is MIXED_INPUT_ENCODING. This page reports K=k(1−v2/v1)².

Q: Is α the included cone angle?

No. α is the wall half-angle from the axis. The included angle is 2α. Enter degrees as alpha_deg, or radians as alpha_rad, not both.

Q: Is this Q = A·v?

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

Q: Where does this run?

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

Compute conical-diffuser loss h = k (v1 − v2)² / (2g) with k = 2.6 sin(α). α is the wall half-angle. Runs locally. Not sudden expansion, not minor-loss K as an input, and not Q=Av.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 17/17 tests · Production surface contract 3/3 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: Conical-diffuser head from v1, v2, and wall half-angle α. Invert v2 or α. Optional Δp from ρ.
Scope: Conical diffuser: h = k (v1 − v2)² / (2g); k = 2.6 sin(α); α wall half-angle; k and K reported, not inputs
Verification: Engine tested · 17/17 tests · Production surface contract 3/3 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 80000-4 — Mechanics — Length, speed, pressure, plane angle
Mott — Applied Fluid Mechanics — Losses in gradual expansions
Sources
Evidence: 2 golden · 10 boundary · 5 property · Production surface contract 3/3 · 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; 163 CURRENT) @ 2026-09-17T13:12:27.589Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

Head lossh = k (v1 − v2)² / (2g)

Diffuser multiplierk = 2.6 sin(α)

Loss coefficientK = k (1 − v2/v1)²

Pressure dropΔp = ρ k (v1 − v2)² / 2

iSI. α is the wall half-angle, entered in degrees as alpha_deg (or radians as alpha_rad). Require 0 < α < arcsin(1/2.6)≈22.62°. k=1 is sudden expansion, not this page. g defaults to 9.80665 m/s². Expansion needs v1 > v2.

How to use

Enter v₁

Upstream mean speed in m/s. URL field is v1_m_s.

Enter v₂

Downstream mean speed in m/s, smaller than v₁. Invert on the v₂ tab.

Enter α

Wall half-angle in degrees. Invert on the α tab. URL field is alpha_deg.

Example calculations

Common configurations with formula and result.

v1=2 m/s, v2=1 m/s, α=7°

Typical diffuser half-angle

k = 2.6 sin(7°)

h≈0.01616 m · k≈0.31686 · K≈0.07922 · Δp≈158.4 Pa at ρ=1000

Invert v₂

h≈0.01616 m, v1=2 m/s, α=7°

v2 = v1 − √(2 g h / k)

v2 = 1 m/s

Gradual Expansion calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 17/17 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: Conical diffuser: h = k (v1 − v2)² / (2g), k = 2.6 sin(α). α is the wall half-angle from the axis. k and K are reported, not inputs. Not sudden expansion.
What it calculates: Conical-diffuser head from v1, v2, and wall half-angle α. Invert v2 or α. Optional Δp from ρ.
Inputs: mode?
v1
v2|h
alpha_deg|alpha_rad
rho?
g?
Outputs: h
v1
v2
k
K
alpha
alpha_deg
g
rho
dP
mode
model
Formula: h = k (v1 − v2)² / (2g), k = 2.6 sin(α)
Assumptions: Conical diffuser: h = k (v1 − v2)² / (2g); k = 2.6 sin(α); α wall half-angle; k and K reported, not inputs
Require 0 < α < arcsin(1/2.6); k=1 is sudden expansion; default g=9.80665 m/s²; default ρ=1000 kg/m³; incompressible; v1 > v2
Not sudden expansion, not minor-loss K given, not Darcy f, not orifice C_d, not Q=Av, not sudden contraction
Units: SI; v in m/s; h in m; α in rad (alpha_deg in °); ρ in kg/m³; Δp in Pa; k and K dimensionless
Boundary conditions: missing v1, v2, h, or α when required → MISSING_REQUIRED_INPUT
v1 ≤ 0, h ≤ 0, or α ≤ 0 → VALUE_MUST_BE_POSITIVE
v2 ≥ v1 or α ≥ arcsin(1/2.6) → VALUE_OUT_OF_RANGE
K, k, C_c, f/Re, Q, C_d → MIXED_INPUT_ENCODING
sudden, contraction, minor, darcy, or orifice as mode → INVALID_MODE
Example: v1=2 v2=1 alpha_deg=7 → h=k/(2g) k=2.6 sin(7°)
Validation cases: 3 published on this page · 17/17 tests · Production surface contract 3/3 · View evidence
v1=2 v2=1 alpha_deg=7 → h=k/(2g) k=2.6 sin(7°)
mode=pressure v1=2 v2=1 alpha_deg=7 → dP=500 k
v1=2 v2=1 alpha_deg=7 K=0.08 → MIXED_INPUT_ENCODING
Sources: ISO 80000-4 — Mechanics — Length, speed, pressure, plane angle
Supports: h is a head; v is a speed; Δp is a pressure; α is a plane angle
Mott — Applied Fluid Mechanics — Losses in gradual expansions
Supports: K=2.6 sin(θ) for a conical enlargement, θ wall half-angle, below the sudden-expansion limit
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute conical-diffuser gradual-expansion loss from the two mean speeds and the wall half-angle.

Supported and not supported

Supported — h from v1, v2, α · invert v2 · invert α · API physics.fluid.gradual_expansion

Not supported — sudden expansion (k=1, no α), catalog minor-loss K, Darcy f, orifice C_d, Q=Av, sudden contraction

Agent / API notes

Capability id: physics.fluid.gradual_expansion · tool id: gradual-expansion · pin 1.0.0.

{ "v1": 2, "v2": 1, "alpha_deg": 7 }

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

Calculator URL stays /calc/physics/gradual-expansion. There is no /calc/fluid.

Other calculators in this family: Sudden expansion, Sudden contraction, Gradual contraction, Minor loss, Flow rate .

}

Frequently asked questions

Key distinctions behind the calculation.

Is this sudden expansion?

No. Borda–Carnot with k=1 and no angle is /calc/physics/sudden-expansion. This page needs α and keeps k<1. Sending k is MIXED_INPUT_ENCODING. α ≥ arcsin(1/2.6) is VALUE_OUT_OF_RANGE.

Is this minor-loss K?