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Physics calculator

Pipe Cross Calculator

Compute equal-area sharp 90° four-way cross loss h = K v_c² / (2g) from dividing or combining flow and branch or through split with equal opposite laterals. Runs locally. Not a 90° tee, not a wye, 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 v_c² / (2g) for equal-area 90° cross; dividing branch (1+q)², dividing run 0.4(1−q)², combining run 1+q²−(q²)/(1+q(0.75+0.25q)²); branch q≤0.5; plus pressure and velocity solves; + O3 mpmath tabulated head.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
h = K v_c² / (2g) for equal-area 90° cross; dividing branch (1+q)², dividing run 0.4(1−q)², combining run 1+q²−(q²)/(1+q(0.75+0.25q)²); branch q≤0.5; 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 catalog K given, not circular-arc R/D, not mitered n, not entrance/exit geometry, not a tee or wye, and not Q=Av.
Numerical scope
O2: equal-area 90° cross K vs a separate-module identity (≤2 ULP). Not catalog K, not circular-arc R/D, not mitered n, not entrance/exit, not a tee/wye, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated pipe-cross vectors. Not catalog K, not circular-arc R/D, not mitered n, not entrance/exit, not a non-90° wye, and not Q=Av.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Equal-area sharp 90° four-way cross head from common-channel speed, dividing or combining pattern, branch or through, and q=Q_path/Q_c with equal opposite laterals. Invert v_c. Optional Δp from ρ.
Scope
Equal-area sharp 90° four-way cross: h = K v_c² / (2g); K referenced to common-channel speed; two opposite laterals are equal; 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 v_c² / (2g) for equal-area 90° cross; dividing branch (1+q)², dividing run 0.4(1−q)², combining run 1+q²−(q²)/(1+q(0.75+0.25q)²); branch q≤0.5; 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 c84e791c3bb3
Legacy regression
7/7 tests · Production surface contract 6/6
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
5/5 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 · Manifest
Sources
  • ISO 80000-4 — Mechanics — Length, speed, pressure
  • Idelchik — Handbook of Hydraulic Resistance — Diagram 7-21 welded equal-area 90° cross; dividing through 7-20 No.1; combining recovery 0.75+0.25q
Sources
Evidence
1 legacy golden · legacy regression suite · 5/5 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 v_c² / (2g)
Dividing branchK_s = (1 + q)²
Dividing runK_st = 0.4 (1 − q)²
Combining branchK_s = 1 + q² − 4(1 − 2q)² / (2 − q)
Combining runK_st = 1 + q² − q² / [1 + q(0.75 + 0.25 q)²]
Pressure dropΔp = ρ K v_c² / 2
iSI. q = Q_path/Q_c. Branch q ≤ 0.5 because the two opposite laterals are equal. v_m_s is the common-channel speed v_c, not the path speed. α is fixed at 90°. g defaults to 9.80665 m/s². Do not send query v — that is calculation_version.

How to use

1

Choose pattern

dividing (split from the common inlet into through plus two laterals) or combining (merge into the common outlet). URL field is kind.

2

Choose path

branch (one 90° lateral) or run (straight through). URL field is path.

3

Enter q and v_c

q_ratio = Q_path/Q_c. Branch q cannot exceed 0.5. v_m_s is the common-channel speed. Invert on the v tab.

Example calculations

Common configurations with formula and result.

ϟ

Dividing branch, q=0.5, v_c=2 m/s

Each opposite lateral takes half the inlet; no through flow

K=(1+q)²=2.25, h=K v_c²/(2g)
h≈0.45887 m · K=2.25 · Δp=4500 Pa at ρ=1000
ϟ

Dividing run, q=0.5

Half the inlet continues through; each lateral takes 0.25

K=0.4(1−q)²=0.1
K=0.1

Pipe Cross calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Equal-area sharp 90° four-way cross. K is referenced to the common-channel speed v_c and reported from the flow pattern and q = Q_path/Q_c. Opposite laterals are equal. Dividing: branch K=(1+q)², run K=0.4(1−q)². Combining: branch K=1+q²−4(1−2q)²/(2−q), run K=1+q²−q²/[1+q(0.75+0.25q)²]. Branch q≤0.5. Not a 90° tee, not a wye, not a catalog fitting K.
What it calculates
Equal-area sharp 90° four-way cross head from common-channel speed, dividing or combining pattern, branch or through, and q=Q_path/Q_c with equal opposite laterals. Invert v_c. Optional Δp from ρ.
Inputs
  • mode?
  • kind?
  • path?
  • q_ratio
  • v_m_s|h
  • rho?
  • g?
Outputs
  • h
  • v
  • K
  • kind
  • path
  • q
  • q_ratio
  • g
  • rho
  • dP
  • alpha_deg
  • mode
  • model
Formula
h = K v_c² / (2g); dividing branch (1+q)²; dividing run 0.4(1−q)²; combining branch 1+q²−4(1−2q)²/(2−q); combining run 1+q²−q²/[1+q(0.75+0.25q)²]
Assumptions
  • Equal-area sharp 90° four-way cross: h = K v_c² / (2g); K referenced to common-channel speed; two opposite laterals are equal; K is reported, not an input
  • Dividing: branch K=(1+q)², run K=0.4(1−q)². Combining: branch K=1+q²−4(1−2q)²/(2−q), run K=1+q²−q²/[1+q(0.75+0.25q)²]. Branch q≤0.5; default g=9.80665 m/s²; default ρ=1000 kg/m³
  • Not a 90° tee, not a non-90° wye, not catalog minor-loss K given, not circular-arc R/D, not mitered n, not pipe-entrance geometry, not pipe-exit K=1, not Q=Av
Units
  • SI; v_c in m/s; h in m; q dimensionless; ρ in kg/m³; Δp in Pa; K dimensionless
Boundary conditions
  • missing v_m_s, h, or q_ratio when required → MISSING_REQUIRED_INPUT
  • v_m_s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • q_ratio > 1, branch q>0.5, or K≤0 on velocity invert → VALUE_OUT_OF_RANGE
  • K, rd, n_miters, θ, α, f, Q, A, v1/v2, ri_d, q_opp, n_legs → MIXED_INPUT_ENCODING
  • flush, circular, perpendicular, colliding, or unknown kind/path/mode → INVALID_MODE
Example
v_m_s=2 kind=dividing path=branch q_ratio=0.5 → h=K v_c²/(2g) K=2.25
Validation cases

3 published on this page · 7/7 tests · Production surface contract 6/6 · View evidence

  • v_m_s=2 kind=dividing path=branch q_ratio=0.5 → h=K v_c²/(2g) K=2.25
  • v_m_s=2 kind=dividing path=run q_ratio=0.5 → K=0.1
  • v_m_s=2 kind=dividing path=branch q_ratio=0.6 → VALUE_OUT_OF_RANGE
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 — Diagram 7-21 welded equal-area 90° cross; dividing through 7-20 No.1; combining recovery 0.75+0.25q
    Supports: Dividing branch K=(1+q)² referenced to v_c; dividing run K=0.4(1−q)²; combining run uses the welded-cross recovery term. Not a 90° tee, not catalog K, not a wye
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute pipe-cross loss from common-channel speed, dividing or combining flow, and the branch or through split of a four-way junction.

Supported and not supported

Supported — h from v_c, kind, path, and q · invert v_c · API physics.fluid.pipe_cross

Not supported — 90° tee, non-90° wye, catalog minor-loss K, circular-arc R/D, mitered n, pipe-entrance geometry, pipe-exit K=1, Darcy f, Q=Av, perpendicular or colliding two-stream custom configs

Agent / API notes

Capability id: physics.fluid.pipe_cross · tool id: pipe-cross · pin 1.0.0.

{ "v_m_s": 2, "kind": "dividing", "path": "branch", "q_ratio": 0.5 }

Modes: head (default), pressure, velocity. Kinds: dividing (default), combining. Paths: branch (default), run. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/pipe-cross. 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, Concentric Reducer Calculator, Critical Depth 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. › → 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. › → 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. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a 90° tee?

No. The equal-area sharp 90° T is /calc/physics/pipe-tee. A tee has one lateral; this cross has two equal opposite laterals. Branch q>0.5 is VALUE_OUT_OF_RANGE.

Is this a wye?

No. A non-90° equal-area wye is /calc/physics/pipe-wye. Sending alpha_deg is MIXED_INPUT_ENCODING.

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 kind, path, and q.

Is this Q = A·v?

No. Volumetric flow is /calc/physics/flow-rate. Sending Q or A is MIXED_INPUT_ENCODING. q_ratio is the dimensionless split Q_path/Q_c.

Where does this run?

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