# Method and assumptions Source: https://meptools.com/tools/duct-pressure-drop-calculator/docs/method/ Nothing here is hidden or fitted. This page states the calculation exactly as implemented, so you can reproduce any reported figure by hand and decide whether the assumptions hold for your case. ## Air properties Density from the ideal gas law, viscosity from Sutherland's law, and the speed of sound from the isentropic relation: ```text ρ = p / (R · T) μ = μ₀ · (T / T₀)^1.5 · (T₀ + S) / (T + S) a = √(γ · R · T) Ma = v / a ``` with `p` in Pa, `T` in K, and: | Constant | Value | | -------------------------------------- | ----------------- | | `R` — specific gas constant, dry air | 287.058 J/(kg·K) | | `γ` — ratio of specific heats | 1.4 | | `μ₀` — reference viscosity | 1.716 × 10⁻⁵ Pa·s | | `T₀` — reference temperature | 273.15 K | | `S` — Sutherland constant | 111 K | | `g` — standard gravity, used for mmH₂O | 9.80665 m/s² | Air is treated as **dry**. Humidity is not an input and is not inferred: at 20 °C and 50% RH the real density is roughly 0.5% below the dry value the tool reports, which is immaterial for a preliminary check but is a real bias, always in the same direction. The Mach number is reported for transparency, not applied — the calculation is **incompressible** throughout. That is the reason for the warning at `Ma ≥ 0.1`. ## Geometry **Circular ducts** are solved directly: ```text A = π · D² / 4 v = Q / A Dₕ = D ``` The friction calculation and the reported velocity both use `D` and `v`, so there is nothing to reconcile. ### Rectangular ducts and the Huebscher equivalent diameter A rectangular duct produces **two** areas and **two** velocities, and the tool keeps them deliberately separate: ```text A = w · h actual area Dₕ = 2 · w · h / (w + h) hydraulic diameter Dₑ = 1.3 · (w · h)^0.625 / (w + h)^0.25 Huebscher equivalent diameter Aₑ = π · Dₑ² / 4 equivalent circular area v = Q / A actual velocity — what you see vₑ = Q / Aₑ equivalent velocity — what friction uses ``` The Huebscher relation gives the diameter of a **circular duct with the same friction loss at the same airflow**. Because `Aₑ` is smaller than `A` for any real rectangular duct, `vₑ` is always higher than `v`. This is not an error: the friction loss is solved in the equivalent circular duct, while the velocity you are shown, judged against the advisory band, and use for velocity pressure is the **real** duct velocity. The equivalent-diameter approach degrades as ducts get flatter, which is what the aspect-ratio warning above 4 is marking. ## Reynolds number and flow regime Two Reynolds numbers are computed, and they are used for different things: ```text Re = ρ · v · Dₕ / μ reported, and classifies the regime Re_friction = ρ · vₑ · Dₑ / μ solves the friction factor ``` For circular ducts they are identical. For rectangular ducts `Re_friction` is the larger of the two. The regime badge is classified on the **reported** Reynolds number: | Regime | Range | | ------------ | ---------------- | | Laminar | Re < 2300 | | Transitional | 2300 ≤ Re < 4000 | | Turbulent | Re ≥ 4000 | ## Friction factor The **Darcy** friction factor (not the Fanning factor, which is one quarter of it), solved on `Re_friction`: ```text Re_friction < 2300: f = 64 / Re_friction otherwise: f = 0.25 / [ log₁₀( ε / (3.7 · Dₑ) + 5.74 / Re_friction^0.9 ) ]² ``` The second form is the **Swamee-Jain** explicit approximation to the Colebrook-White equation. It is an approximation with no iteration, valid to roughly ±1% against Colebrook over `10⁻⁶ ≤ ε/D ≤ 10⁻²` and `5000 ≤ Re ≤ 10⁸` — comfortably inside the range of ordinary ventilation ductwork, and outside it in the transitional band, which is exactly what that warning marks. There is no explicit correlation in 2300–4000; the turbulent form is applied and flagged rather than refused. `ε` is the absolute roughness in metres — the preset value from card 03, converted from mm. ## Pressure drop ```text R = f · ρ · vₑ² / (2 · Dₑ) friction rate, Pa/m Δp_straight = R · L Pa q = ½ · ρ · v² velocity pressure, Pa (actual velocity) Δp_fittings = Σ K · q Pa Δp_total = Δp_straight + Δp_fittings Pa Δp (mmH₂O) = Δp_total / 9.80665 ``` Written as one expression, this is the Darcy-Weisbach equation with a fitting term: ```text Δp = f · (L / Dₑ) · ½ρvₑ² + Σ K · ½ρv² ``` Note which velocity sits in each term. The friction term runs on the **equivalent** velocity and diameter; the fitting term runs on the **actual** velocity pressure. For a circular duct the two collapse into the familiar `Δp = (f·L/D + Σ K) · ½ρv²`. For a rectangular duct they do not, and a hand-check that uses the actual velocity in the friction term will not reproduce the tool's number. ## Assumptions and limits, stated plainly - **One run, one size.** A single duct of constant cross-section. No networks, no branches carrying different flows, no index-run selection. - **Steady, incompressible, isothermal, fully-developed flow.** No heat transfer, no density change along the run, no entry-length effects. - **Dry air.** Humidity is neither an input nor inferred. - **Static regain and dynamic-pressure recovery are ignored.** Σ K losses are additive with no credit for velocity-pressure recovery at expansions. - **Rigid duct only.** There is no flexible-duct correction; a flexible run's friction is substantially higher than the equivalent rigid duct and is not modelled. - **No leakage, no elevation term, no fan curve.** Air in equals air out, and the result is a loss across the run, not a system operating point. - **K values are referenced to this run's actual velocity pressure.** A published K referenced to another velocity must be converted before entry. - **The friction factor is an approximation**, and in the transitional band an acknowledged poor one. ## Tolerance The calculation kernel is covered by unit tests that pin the worked results to within ±0.5% of independently computed reference values. That tolerance describes the **implementation**, not the physics: it says the code solves these equations correctly, not that these equations predict your duct.