Verified calculation capability

Pipe bend Calculation Capability

Circular-arc bend h=K v²/(2g), K=0.21 (θ/90)/√(R/D). Not minor-loss K given, not mitered, not Darcy, not Q=Av. · Physics · Fluid

Capability ID physics.fluid.pipe_bend · version 1.0.0

Open calculator REST API MCP Evidence Evidence JSON Review policy

Model

Head loss

Head loss: h = K v² / (2g)
Loss coefficient: K = 0.21 (θ/90) / √(R/D)
Pressure drop: Δp = ρ K v² / 2
Speed invert: v = √(2 g h / K)

Solve modes: head · pressure · velocity · angle

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`v`	Mean speed in m/s. Required except velocity invert. Alternative: speed. Do not send query v (that is calculation_version).	m/s	input
`speed`	Mean speed in m/s (REST alias of v_m_s).	—	input
`theta_deg`	Bend deflection in degrees. Required except angle invert. Alternative: theta_rad. Do not send both. Range (0, 180].	—	input / output
`theta_rad`	Bend deflection in radians. Alternative: theta_deg. Do not send both.	—	input
`rd`	Centerline radius over diameter R/D. Required. Alternatives: r_d, R_D, r_over_d. Require R/D ≥ 1.	—	input / output
`r_d`	R/D alias of rd.	—	input
`h`	Head loss in m. Required for velocity and angle invert. Alternatives: hm, h_m.	—	input / output
`h`	Head loss in m (query/UI). REST may send h.	m	input
`rho`	Density in kg/m³. Optional; default 1000.	—	input / output
`g`	Gravity in m/s². Optional; default 9.80665.	—	input / output
`v`	velocity	—	output
`K`	—	—	output
`theta`	—	—	output
`dP`	—	—	output
`model`	—	—	output

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

Circular-arc bend: h = K v² / (2g); K = 0.21 (θ/90)/√(R/D); Idelchik 90° B1 scaled linearly in θ; K is reported, not an input
Require 0 < θ ≤ 180° and R/D ≥ 1; default g=9.80665 m/s²; default ρ=1000 kg/m³; incompressible
Not minor-loss K given, not mitered R/D<1, not Darcy f, not orifice C_d, not Q=Av, not gradual contraction α

Not supported

catalog minor-loss K
mitered R/D<1
Darcy f
gradual-contraction α
Q=Av

Example

One numerical check. The calculator page holds worked steps and FAQ.

v_m_s=2 theta_deg=90 rd=2 → h=K v²/(2g) K=0.21/√2

Open this example in the calculator

Trust

Review-policy: engine tested — automated tests, sources, version lock, and evidence published. Supplemental to CVP; not Domain Expert Review. See CVP and Review policy.

Tier: Engine tested — catalog token engine_tested
Calculation version: 1.0.0
Evidence tests: 16/16 calculation tests · golden 2/2 · boundary 9/9 · property 5/5
Engine tested: Yes
Source checked: Yes
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.0 (workspace artifacts agree)

Production attestation: 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) @ 2026-09-17T15:07:30.016Z
Semantic contract: PASS
Evidence attested at: 2026-09-17T15:07:29.295Z
Last production check: 2026-09-17T15:07:29.295Z
Public / cache: PASS
Origin: PASS

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 3/3.

Tool id: pipe-bend
Engine: pipe-bend
REST: POST /api/v1/calc/pipe-bend
MCP: execute_calculation with capability_id: "physics.fluid.pipe_bend"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.0)
Versioned output schema: Immutable pin (1.0.0)

References

Human-readable sources plus CVP reference evidence classes.

Sources

ISO 80000-4 — Mechanics — Length, speed, pressure, plane angle
Idelchik — Handbook of Hydraulic Resistance — Circular-arc bends, B1 at 90°

Try with an agent

Inspect this capability via same-origin MCP, then call execute_calculation.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"physics.fluid.pipe_bend"}}}'

Inspect the trust tier and required assurance level before execution. Pin calculation_version (1.0.0) when reproducibility matters. Full protocol notes: MCP docs.