CVP 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 CVP VERIFIED CURRENT Engineering

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)

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	m/s	input
`speed`	Mean speed in m/s	—	input
`theta_deg`	Bend deflection in degrees	—	input / output
`theta_rad`	Bend deflection in radians	—	input
`rd`	Centerline radius over diameter R/D	—	input / output
`r_d`	R/D alias of rd	—	input
`h`	Head loss in m	—	input / output
`h`	Head loss in m	m	input
`rho`	Density in kg/m³	—	input / output
`g`	Gravity in m/s²	—	input / output
`v`	—	—	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

CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · h = K v² / (2g) with K = 0.21 (θ/90)/√(R/D), plus pressure, velocity, and angle solves, + O3 mpmath tabulated head.. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · h = K v² / (2g) with K = 0.21 (θ/90)/√(R/D), plus pressure, velocity, and angle solves, + O3 mpmath tabulated head.· View Manifest · CVP overview · Specification
Verified scope: h = K v² / (2g) with K = 0.21 (θ/90)/√(R/D), plus pressure, velocity, and angle solves, + O3 mpmath tabulated head.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (head, solve, alias, awkward, invalid-domain)
Deferred: Not minor-loss K given, not mitered, not Darcy, and not Q=Av.
Calculation version: 1.0.0
Verification: 16/16 calculation tests · 2/2 legacy golden · 9/9 legacy boundary · 5/5 legacy property
Production surface contract: 5/5
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 4fe81755cb43
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Numerical scope: O2: circular-arc bend K=0.21 (θ/90)/√(R/D) vs a separate-module identity (≤2 ULP). Not minor-loss K given, not mitered, not Darcy, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated pipe-bend vectors. Not minor-loss K given, not mitered, not Darcy, and not Q=Av.
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.
CVP suite: 4/4 golden · 8/8 CVP boundary · 9/9 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 5/5 CVP contract
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)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

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 5/5.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/pipe-bend' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"v_m_s":2,"theta_deg":90,"rd":2},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "v_m_s": 2,
    "theta_deg": 90,
    "rd": 2
  },
  "calculation_version": "1.0.0"
}

POST /api/v1/calc/pipe-bend. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

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°

Reference evidence classes: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

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"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

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: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"physics.fluid.pipe_bend","inputs":{"v_m_s":2,"theta_deg":90,"rd":2},"calculation_version":"1.0.0"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.0.0) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.