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Physics calculator

Pipe Bend Calculator

Compute circular-arc bend loss h = K v² / (2g) with K = 0.21 (θ/90)/√(R/D). Runs locally. Not minor-loss K as an input, not mitered, not Darcy, 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 v² / (2g) with K = 0.21 (θ/90)/√(R/D), plus pressure, velocity, and angle solves, + O3 mpmath tabulated head.
Input interpretation
Enter values to calculate.
Result
—
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) · Matrix
Deferred
Not minor-loss K given, not mitered, not Darcy, and not Q=Av.
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.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Circular-arc bend head from mean speed, deflection θ, and R/D. Invert v or θ. Optional Δp from ρ.
Scope
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
Verification
Engine tested · Source checked · v1.0.0 · 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
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
Legacy regression
16/16 tests · Production surface contract 5/5
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
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 · Manifest
Sources
  • ISO 80000-4 — Mechanics — Length, speed, pressure, plane angle
  • Idelchik — Handbook of Hydraulic Resistance — Circular-arc bends, B1 at 90°
Sources
Evidence
2 legacy golden · 9 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 9/9 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 v² / (2g)
Loss coefficientK = 0.21 (θ/90) / √(R/D)
Pressure dropΔp = ρ K v² / 2
Speed invertv = √(2 g h / K)
iSI. θ is the included deflection, entered in degrees as theta_deg (or radians as theta_rad). Require 0 < θ ≤ 180° and R/D ≥ 1. R/D < 1 is mitered, not this page. g defaults to 9.80665 m/s². Do not send query v — that is calculation_version; use v_m_s.

How to use

1

Enter v

Mean speed in the pipe in m/s. URL field is v_m_s. Invert on the v tab.

2

Enter θ

Bend deflection in degrees. Invert on the θ tab. URL field is theta_deg.

3

Enter R/D

Centerline radius over diameter. URL field is rd. Must be ≥ 1.

Example calculations

Common configurations with formula and result.

ϟ

v=2 m/s, θ=90°, R/D=2

Smooth 90° circular-arc

K = 0.21 / √2
h≈0.03028 m · K≈0.1485 · Δp≈297.0 Pa at ρ=1000
ϟ

Invert v

h≈0.03028 m, θ=90°, R/D=2

v = √(2 g h / K)
v = 2 m/s

Pipe Bend calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Circular-arc pipe bend: h = K v² / (2g), K = 0.21 (θ/90)/√(R/D). That is Idelchik 90° B1 scaled linearly with deflection. K is reported, not an input. Not a catalog fitting K.
What it calculates
Circular-arc bend head from mean speed, deflection θ, and R/D. Invert v or θ. Optional Δp from ρ.
Inputs
  • mode?
  • v_m_s|h
  • theta_deg|theta_rad
  • rd
  • rho?
  • g?
Outputs
  • h
  • v
  • K
  • theta
  • theta_deg
  • rd
  • g
  • rho
  • dP
  • mode
  • model
Formula
h = K v² / (2g), K = 0.21 (θ/90) / √(R/D)
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
Units
  • SI; v in m/s; h in m; θ in rad (theta_deg in °); rd dimensionless; ρ in kg/m³; Δp in Pa; K dimensionless
Boundary conditions
  • missing v_m_s, h, θ, or rd when required → MISSING_REQUIRED_INPUT
  • v_m_s ≤ 0, h ≤ 0, θ ≤ 0, or rd ≤ 0 → VALUE_MUST_BE_POSITIVE
  • rd < 1 or θ > 180° → VALUE_OUT_OF_RANGE
  • K, k, f/Re, Q, C_d, v1/v2, alpha_deg → MIXED_INPUT_ENCODING
  • minor, mitered, darcy, or orifice as mode → INVALID_MODE
Example
v_m_s=2 theta_deg=90 rd=2 → h=K v²/(2g) K=0.21/√2
Validation cases

3 published on this page · 16/16 tests · Production surface contract 5/5 · View evidence

  • v_m_s=2 theta_deg=90 rd=2 → h=K v²/(2g) K=0.21/√2
  • mode=pressure v_m_s=2 theta_deg=90 rd=2 → dP=1000 K v²/2
  • v_m_s=2 theta_deg=90 rd=2 K=0.15 → 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; θ is a plane angle
  • Idelchik — Handbook of Hydraulic Resistance — Circular-arc bends, B1 at 90°
    Supports: ζ_90 = 0.21 / √(R/D) for a circular-arc of circular section in the quadratic region; this page scales linearly with θ/90°
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute circular-arc elbow loss from mean speed, deflection, and curvature ratio.

Supported and not supported

Supported — h from v, θ, R/D · invert v · invert θ · API physics.fluid.pipe_bend

Not supported — catalog minor-loss K, mitered R/D<1, Darcy f, gradual-contraction α, Q=Av

Agent / API notes

Capability id: physics.fluid.pipe_bend · tool id: pipe-bend · pin 1.0.0.

{ "v_m_s": 2, "theta_deg": 90, "rd": 2 }

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/pipe-bend. 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 minor-loss K?

No. Catalog fitting K is /calc/physics/minor-loss. Sending K is MIXED_INPUT_ENCODING. This page reports K=0.21 (θ/90)/√(R/D).

Is this a mitered elbow?

No. Circular-arc needs R/D ≥ 1. R/D < 1 is VALUE_OUT_OF_RANGE. This page does not take a miter count or Crane f_T.

Is this Darcy–Weisbach?

No. Pipe friction from a given f is /calc/physics/darcy-weisbach. Sending f, L, D, or Re is MIXED_INPUT_ENCODING. Geometry is rd = R/D, not a separate D.

Is this gradual contraction?

No. Wall half-angle α is /calc/physics/gradual-contraction. Sending alpha_deg is MIXED_INPUT_ENCODING. This page takes deflection θ and R/D.

Is this Q = A·v?

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

Where does this run?

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