Physics calculator

Pipe Bend Calculator

Q: 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).

Q: 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.

Q: 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.

Q: 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.

Q: Is this Q = A·v?

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

Q: Where does this run?

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

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 Engine tested · Source checked · 16/16 tests · Production surface contract 3/3 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 16/16 tests · Production surface contract 3/3 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 80000-4 — Mechanics — Length, speed, pressure, plane angle
Idelchik — Handbook of Hydraulic Resistance — Circular-arc bends, B1 at 90°
Sources
Evidence: 2 golden · 9 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production: 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

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

Enter v

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

Enter θ

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

Enter R/D

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

Example calculations

Common configurations with formula and result.

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

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

Engine tested 16/16 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.0

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 3/3 · 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: Minor loss, Mitered bend, Pipe entrance, Gradual contraction, Sudden expansion, Darcy–Weisbach .

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Frequently asked questions

Key distinctions behind the calculation.

Is this minor-loss K?