CVP verified calculation capability

Elbow flow meter Calculation Capability

90° elbow tap meter Q=C A √(2Δp/ρ) with C=√((R/D)/2). Not pipe-bend K, not Pitot C=1, not venturi, not ISO 5167-6 wedge, not Q=Av, not Darcy.

Physics · Fluid Capability ID physics.fluid.elbow_meter version 1.0.0 CVP VERIFIED CURRENT Engineering

Model

Discharge

Discharge: Q = C A √(2Δp / ρ)
Elbow C: C = √((R/D) / 2)
Area: A = (π/4) D²

Modes: flow · pressure · coefficient

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`D`	Pipe inside diameter in m	—	input / output
`R`	Bend centerline radius in m	—	input / output
`rd`	Centerline R/D	—	input / output
`dP`	Inner-to-outer tap differential pressure in Pa	—	input / output
`rho`	Density in kg/m³	—	input / output
`Q`	Volume flow in m³/s	—	input / output
`A`	—	—	output
`C`	—	—	output
`Cd`	—	—	output
`v`	—	—	output
`model`	—	—	output

Assumptions / Validity

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

Assumptions

90° elbow tap meter: Q = C A √(2Δp/ρ); C=√((R/D)/2); A=(π/4)D²; inner/outer radius taps; incompressible; C is reported, not an input
R/D∈[1.0, 3.0]; D∈[25 mm, 600 mm]; 90° only; C from R/D, not a given Pitot C=1
Not pipe-bend K=0.21 (θ/90)/√(R/D), not Pitot v=C√(2Δp/ρ), not venturi, not ISO 5167-6 wedge, not Q=Av, not Darcy f

Not supported

pipe-bend K
Pitot C=1
venturi
ISO 5167-6 wedge
given C_d
Q=Av
Darcy f
variable θ

Example

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

D=0.1 R=0.15 dP=10000 rho=1000 → C≈0.86603 Q≈0.030418

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Q = C A √(2Δp/ρ) with C=√((R/D)/2), A=(π/4)D²; plus pressure and coefficient solves; + O3 mpmath tabulated discharge.. 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 · Q = C A √(2Δp/ρ) with C=√((R/D)/2), A=(π/4)D²; plus pressure and coefficient solves; + O3 mpmath tabulated discharge.· View Manifest · CVP overview · Specification
Verified scope: Q = C A √(2Δp/ρ) with C=√((R/D)/2), A=(π/4)D²; plus pressure and coefficient solves; + O3 mpmath tabulated discharge.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (flow, solve, alias, awkward, invalid-domain)
Deferred: Not pipe-bend K, not Pitot C=1, not venturi/orifice β, not Q=Av, and not Darcy.
Calculation version: 1.0.0
Verification: 6/6 calculation tests · 1/1 legacy golden · 0/0 legacy boundary · 5/5 legacy property
Production surface contract: 6/6
Versions: Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-27.o2-o3
Verification revision: 2026-09-27.o2-o3 · 1/1 property · digest 958679d30c6a
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Numerical scope: O2: elbow meter Q=C A √(2Δp/ρ), C=√((R/D)/2) vs a separate-module identity (≤2 ULP). Not pipe-bend K, not Pitot C=1, not venturi/orifice, not Q=Av, and not Darcy. ≤2 ULP vs O3 applies only to the published tabulated elbow-meter vectors. Not pipe-bend K, not Pitot C=1, not venturi/orifice, 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 · 1/1 CVP boundary · 2/2 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 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 6/6.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/elbow-meter' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"D":0.1,"R":0.15,"dP":10000,"rho":1000},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "D": 0.1,
    "R": 0.15,
    "dP": 10000,
    "rho": 1000
  },
  "calculation_version": "1.0.0"
}

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

Full REST contract: REST API docs.

Tool id: elbow-meter
Engine: elbow-meter
REST: POST /api/v1/calc/elbow-meter
MCP: execute_calculation with capability_id: "physics.fluid.elbow_meter"
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
Elbow tap meter — centrifugal Δp — C=√((R/D)/2) from inner/outer taps

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.elbow_meter"},"_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.elbow_meter","inputs":{"D":0.1,"R":0.15,"dP":10000,"rho":1000},"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.