How it works — three pressures, one Bernoulli chain
Air in a duct carries pressure in two wallets: static \(p_s\) (pushes on the duct wall, does the work) and velocity \(p_v\) (rides in the moving air, recoverable only by decelerating it). Their sum is the total pressure \(p_t\). A fan is sized by accounting for every dollar spent along the path.
1 · The three pressure definitions
The US shortcut folds ρ = 0.0752 lb/ft³ and the fpm→in-wg conversion into the constant 4005 — e.g. 1,000 fpm → (1000/4005)² ≈ 0.062 in-wg. Bernoulli with losses: total pressure drops only where friction or fittings dissipate energy; static pressure additionally shifts whenever the velocity changes (area change) — acceleration buys \(p_v\) with \(p_s\), deceleration sells it back.
2 · Fan total pressure and fan static pressure
FTP is the total-pressure rise the fan must produce between inlet and outlet planes. FSP removes the outlet velocity pressure \(p_{v,o}\): with a free discharge, that fraction of the fan's work stays locked in the moving air and never becomes useful static. The classic static method (all static losses + the exit velocity pressure) is conservative — it equals FTP and ignores exactly this \(p_{v,o}\) credit, overpredicting the requirement by \(p_{v,o}\).
3 · Step-by-step build-up along the path
Each section is evaluated at its own velocity pressure: Darcy friction with a Colebrook λ for galvanized steel (ε ≈ 0.15 mm, standard air), plus the sum of fitting coefficients ζ. Walking the path from ambient (gauge 0) through the suction side, across the fan, and out to the free discharge closes the chain:
$$\text{FTP} \;=\; \sum_{\text{all sections}} \Delta p \;+\; p_{v,\text{exit}} \qquad\Longrightarrow\qquad \text{FSP} \;=\; \sum_{\text{all sections}} \Delta p \;+\; p_{v,\text{exit}} - p_{v,o}$$4 · Simplified fitting ζ table (reusable)
| Fitting (simplified) | ζ — multiples of the local pv |
|---|---|
| 45° elbow | 0.12 |
| 90° smooth elbow, r/D = 1.5 | 0.15 |
| 90° elbow, r/D = 1.0 (5-gore) | 0.22 |
| 90° mitered elbow with turning vanes | 0.35 |
| 90° mitered elbow, no vanes | 1.20 |
| Tee — branch flow | 1.00 |
| Sharp-edged entrance (duct from plenum) | 0.50 |
| Butterfly damper ≈ ⅔ open | 1.00 |
| Sudden enlargement, area ratio 2:1 (refer to small-duct pv) | 0.25 |
| Sudden contraction, area ratio 2:1 | 0.15 |
These are engineering estimates for preliminary build-ups — final selections belong on manufacturer data. Fittings in series add: two r/D = 1.0 elbows enter Σζ = 0.44.
5 · Traffic-light thresholds
| Margin (available − required) / required | Verdict | Action |
|---|---|---|
| 🟢 −5% … +15% | Matched | None — healthy, non-wasteful reserve |
| 🟡 +15% … +40% | Oversized | Balance with dampers or trim speed (VFD) |
| 🔴 > +40% | Badly oversized | Re-select the fan or slow it down |
| 🟡 −15% … −5% | Slight shortfall | Check filter loading, coil fouling, leakage |
| 🔴 < −15% | Undersized | Airflow falls far below design — re-select |
| Exit kinetic share pv,exit / FTP | Verdict |
|---|---|
| 🟢 ≤ 10% | Little fan work thrown away at discharge |
| 🟡 10% … 20% | Notable — consider a larger end duct or exit cone |
| 🔴 > 20% | Dominates — enlarge the final section or add a diffuser |
Quick reference — standard air: CFM ↔ fpm ↔ in-wg
Round galvanized duct at three common target velocities. Velocity pressure at those velocities (standard air): 1,000 fpm → 0.062 · 1,500 fpm → 0.140 · 2,000 fpm → 0.249 · 2,500 fpm → 0.390 in-wg.
| D (in) | A (ft²) | cfm @ 1,000 fpm | cfm @ 1,500 fpm | cfm @ 2,000 fpm | cfm @ 2,500 fpm |
|---|---|---|---|---|---|
| 8 | 0.349 | 349 | 524 | 698 | 873 |
| 10 | 0.545 | 545 | 818 | 1,091 | 1,364 |
| 12 | 0.785 | 785 | 1,178 | 1,571 | 1,963 |
| 14 | 1.069 | 1,069 | 1,604 | 2,138 | 2,673 |
| 16 | 1.396 | 1,396 | 2,094 | 2,793 | 3,491 |
| 18 | 1.767 | 1,767 | 2,651 | 3,534 | 4,418 |
| 20 | 2.182 | 2,182 | 3,272 | 4,363 | 5,454 |
| 22 | 2.640 | 2,640 | 3,960 | 5,280 | 6,600 |
| 24 | 3.142 | 3,142 | 4,712 | 6,283 | 7,854 |
| 28 | 4.276 | 4,276 | 6,414 | 8,552 | 10,690 |
| 32 | 5.585 | 5,585 | 8,378 | 11,170 | 13,963 |
Same table in SI (v in m/s, Q in m³/h). Velocity pressure: 4 m/s → 9.6 · 6 m/s → 21.7 · 8 m/s → 38.5 · 10 m/s → 60.2 Pa.
| D (mm) | A (m²) | m³/h @ 4 m/s | m³/h @ 6 m/s | m³/h @ 8 m/s | m³/h @ 10 m/s |
|---|---|---|---|---|---|
| 250 | 0.049 | 707 | 1,060 | 1,414 | 1,767 |
| 315 | 0.078 | 1,122 | 1,683 | 2,244 | 2,806 |
| 400 | 0.126 | 1,810 | 2,714 | 3,619 | 4,524 |
| 500 | 0.196 | 2,827 | 4,241 | 5,655 | 7,069 |
| 630 | 0.312 | 4,489 | 6,733 | 8,978 | 11,222 |
| 800 | 0.503 | 7,238 | 10,857 | 14,476 | 18,096 |
| 1000 | 0.785 | 11,310 | 16,965 | 22,619 | 28,274 |
Symbol table
| Symbol | Meaning | SI | HVAC-US |
|---|---|---|---|
| Q | airflow | m³/s | cfm |
| v, V | duct air velocity | m/s | fpm |
| A | duct area, πD²/4 | m² | ft² |
| ρ | air density (standard air 1.2041) | kg/m³ | lb/ft³ |
| pv (Pd) | velocity (dynamic) pressure, ½ρv² | Pa | in-wg |
| ps | static pressure | Pa | in-wg |
| pt | total pressure, ps + pv | Pa | in-wg |
| pt,i, pt,o | fan inlet / outlet total pressure | Pa | in-wg |
| FTP | fan total pressure, pt,o − pt,i | Pa | in-wg |
| FSP | fan static pressure, FTP − pv,o | Pa | in-wg |
| pv,o | fan outlet velocity pressure | Pa | in-wg |
| λ | Darcy friction factor (Colebrook, ε ≈ 0.15 mm) | — | — |
| ζ | fitting loss coefficient, × local pv | — | — |
| L, D | section length / diameter | m | ft, in |
Worked anchor — one closed loop (the defaults)
6,000 cfm of standard air, sections as preset (30 ft + 50 ft of 24 in, 40 ft + 30 ft of 20 in, ζ = 0.50 / 0.44 / 0.25 / 0.35), available fan FSP 1.30 in-wg. Every step below reproduces in the calculator; values rounded to 0.001 in-wg (the tool keeps full precision, so re-adding the rounded steps drifts at most ±0.001).
| Step | Quantity | Value |
|---|---|---|
| 1 | 24 in duct area A = πD²/4 | 3.142 ft² |
| 2 | velocity v = 6,000 / 3.142 | 1,910 fpm |
| 3 | velocity pressure pv = ρv²/2 (24 in sections) | 0.228 in-wg |
| 4 | 20 in duct → v = 2,750 fpm → pv | 0.472 in-wg |
| 5 | Suction 1: 30 ft friction + ζ 0.50 · pv | 0.055 + 0.114 = 0.169 |
| 6 | Supply 1: 50 ft friction + ζ 0.44 · pv | 0.092 + 0.100 = 0.192 |
| 7 | Supply 2: 40 ft friction + ζ 0.25 · pv | 0.185 + 0.118 = 0.303 |
| 8 | Supply 3: 30 ft friction + ζ 0.35 · pv | 0.139 + 0.165 = 0.304 |
| 9 | fan inlet total pt,i | −0.169 in-wg |
| 10 | fan outlet total pt,o = exit pv + supply losses | 1.270 in-wg |
| 11 | FTP = 1.270 − (−0.169) | 1.439 in-wg |
| 12 | FSP = 1.439 − 0.228 | 1.212 in-wg |
| 13 | margin vs available 1.30 in-wg → (1.30 − 1.212)/1.212 | 🟢 +7.3% — matched |
| 14 | exit kinetic share 0.472 / 1.439 | 🔴 32.8% — enlarge the end duct |
The loop closes at full precision (4 decimals): FSP + pv,o = 1.2118 + 0.2275 = 1.4393 = FTP = Σ section losses (0.9675) + exit pv (0.4718). The 🔴 exit share is real teaching, not an error — 2,750 fpm at a free discharge burns nearly a third of FTP as jet kinetic energy.