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Mitered Bend Calculator

Compute sharp mitered-elbow loss h = K v² / (2g) with K = n [sin²(δ/2) + 2 sin⁴(δ/2)], δ = θ/n. Runs locally. Not circular-arc R/D, not minor-loss K as an input, not Crane f_T, and not Q=Av.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Source checked · 15/15 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Sharp mitered-elbow head from mean speed, total deflection θ, and miter count n. Invert v or θ. Optional Δp from ρ.
Scope
Sharp segmented miter: h = K v² / (2g); K = n [sin²(δ/2) + 2 sin⁴(δ/2)], δ = θ/n; K is reported, not an input
Verification
Engine tested · 15/15 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 — Sharp-edged / mitered elbows
Sources
Evidence
2 golden · 8 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 = n [sin²(δ/2) + 2 sin⁴(δ/2)], δ = θ/n
Pressure dropΔp = ρ K v² / 2
Speed invertv = √(2 g h / K)
iSI. θ is total deflection, entered in degrees as theta_deg (or radians as theta_rad). n_miters is the number of miter welds (positive integer). Each miter δ = θ/n must be ≤ 90°, so n=1 allows θ ≤ 90° only. 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

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

3

Enter n

Number of miter welds. URL field is n_miters. Must be a positive integer.

Example calculations

Common configurations with formula and result.

ϟ

v=2 m/s, θ=90°, n=1

Single 90° sharp miter

K = sin²(45°) + 2 sin⁴(45°) = 1
h≈0.20394 m · K=1 · Δp=2000 Pa at ρ=1000
ϟ

Invert v

h≈0.20394 m, θ=90°, n=1

v = √(2 g h / K)
v = 2 m/s

Mitered Bend calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Sharp segmented miter: h = K v² / (2g), K = n [sin²(δ/2) + 2 sin⁴(δ/2)], δ = θ/n. K is reported, not an input. Not a catalog fitting K and not a circular-arc R/D.
What it calculates
Sharp mitered-elbow head from mean speed, total deflection θ, and miter count n. Invert v or θ. Optional Δp from ρ.
Inputs
  • mode?
  • v_m_s|h
  • theta_deg|theta_rad
  • n_miters
  • rho?
  • g?
Outputs
  • h
  • v
  • K
  • theta
  • theta_deg
  • n
  • g
  • rho
  • dP
  • mode
  • model
Formula
h = K v² / (2g), K = n [sin²(δ/2) + 2 sin⁴(δ/2)], δ = θ/n
Assumptions
  • Sharp segmented miter: h = K v² / (2g); K = n [sin²(δ/2) + 2 sin⁴(δ/2)], δ = θ/n; K is reported, not an input
  • Require n ≥ 1 integer, 0 < θ ≤ 180°, and each miter δ ≤ 90° (n=1 therefore θ ≤ 90°); default g=9.80665 m/s²; default ρ=1000 kg/m³; incompressible
  • Not circular-arc R/D, not minor-loss K given, not Crane f_T, not Darcy f, not orifice C_d, not Q=Av
Units
  • SI; v in m/s; h in m; θ in rad (theta_deg in °); n dimensionless integer; ρ in kg/m³; Δp in Pa; K dimensionless
Boundary conditions
  • missing v_m_s, h, θ, or n_miters when required → MISSING_REQUIRED_INPUT
  • v_m_s ≤ 0, h ≤ 0, θ ≤ 0, or n_miters ≤ 0 → VALUE_MUST_BE_POSITIVE
  • non-integer n, each miter > 90°, or θ > 180° → VALUE_OUT_OF_RANGE
  • K, k, rd, f/f_T/Re, Q, C_d, v1/v2, alpha_deg → MIXED_INPUT_ENCODING
  • circular, minor, darcy, or orifice as mode → INVALID_MODE
Example
v_m_s=2 theta_deg=90 n_miters=1 → h=K v²/(2g) K=1
Validation cases

3 published on this page · 15/15 tests · Production surface contract 3/3 · View evidence

  • v_m_s=2 theta_deg=90 n_miters=1 → h=K v²/(2g) K=1
  • mode=pressure v_m_s=2 theta_deg=90 n_miters=1 → dP=1000 K v²/2
  • v_m_s=2 theta_deg=90 n_miters=1 K=0.9 → 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 — Sharp-edged / mitered elbows
    Supports: Preliminary sharp-miter estimate ζ = sin²(δ/2) + 2 sin⁴(δ/2) per miter; n equal miters sum. Not circular-arc B1, not Crane f_T
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute sharp mitered-elbow loss from mean speed, total deflection, and miter count.

Supported and not supported

Supported — h from v, θ, n · invert v · invert θ · API physics.fluid.mitered_bend

Not supported — circular-arc R/D, catalog minor-loss K, Crane f_T, Darcy f, Q=Av

Agent / API notes

Capability id: physics.fluid.mitered_bend · tool id: mitered-bend · pin 1.0.0.

{ "v_m_s": 2, "theta_deg": 90, "n_miters": 1 }

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/mitered-bend. There is no /calc/fluid.

Other calculators in this family: Pipe bend, Pipe entrance, Minor loss, Sudden expansion, Darcy–Weisbach .

h Pipe bend › h Pipe entrance › K Minor loss › h Sudden expansion › hf Darcy–Weisbach ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this circular-arc pipe-bend?

No. Smooth R/D ≥ 1 is /calc/physics/pipe-bend. Sending rd is MIXED_INPUT_ENCODING. This page takes n_miters, not R/D.

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 θ and n.

Is this Crane f_T?

No. Crane mitre tables as multiples of f_T are Darcy-adjacent. Sending f, f_T, L, D, or Re is MIXED_INPUT_ENCODING.

Is this Darcy–Weisbach?

No. Pipe friction from a given f is /calc/physics/darcy-weisbach.

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 miteredBend wrapper.