Engineering Calculators

P_hyd = ρ × g × Q × H / (3,600 × 1,000)  [kW]
P_shaft = P_hyd / η_pump
P_motor = P_shaft / η_motor
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ρ = 1,000 kg/m³  |  g = 9.81 m/s²
Typical η_pump = 70–85%  |  η_motor = 92–96%

NPSHa = (P_atm − P_v)/(ρg) + Z_s − h_fs
h_fs = 10.67 × L × Q^1.852 / (C^1.852 × D^4.87)  [H-W]
P_atm = 101,325 × (1 − 2.256×10⁻⁵ × Alt)^5.256  [Pa]
P_vapor: 610.78 × exp(17.27T/(T+237.3))  [Pa]
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Z_s + → flooded suction  |  Z_s − → suction lift
NPSHa must exceed pump NPSHr by ≥ 0.5 m (ISO 9906)

Pump curve:  H = H₀ − a·Q²  →  a = (H₀ − H_r) / Q_r²
System curve: H = H_stat + R·Q² →  R = h_f / Q_design²
Operating point: Q* = √[ (H₀ − H_stat) / (a + R) ]
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Duty should sit near the pump's BEP (best-efficiency point).
Far-left of BEP → recirculation; far-right → cavitation / overload.
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Design flow only calibrates the system curve (sets R). The duty point is the
curve intersection and can land beyond design flow if the pump is oversized.

P_input = ρ·g·Q·H / (3.6×10⁶ · η_p · η_m) [kW]
Specific energy e = P_input / Q [kWh/m³]
Annual cost = P_input × hours × days × tariff
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Typical e: 0.3–0.5 kWh/m³ (low lift) · 0.5–1.5 (high lift / long mains)

Q₂/Q₁ = N₂/N₁
H₂/H₁ = (N₂/N₁)²
P₂/P₁ = (N₂/N₁)³
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Power is cubic — reducing speed 30% saves ~66% energy
Valid for centrifugal pumps (not positive displacement)

Nq = N × √Q / H^(3/4)  [metric, rpm · m³/s · m]
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Nq < 20 → Radial centrifugal (high head)
Nq 20–60 → Mixed flow
Nq > 60 → Axial / propeller (high Q, low head)
Dimensionless: Ns = ω√Q / (gH)^(3/4)

hf = 10.67 × L × Q^1.852 / (C^1.852 × D^4.87)
V = Q / A = 4Q / (π × D²)
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Q in m³/s  |  D in m  |  hf in m
Valid for turbulent flow (Re > 4,000)
Typical velocity range: 0.6 – 3.0 m/s

V = Q / A = 4Q / (π × D²)
Re = V × D / ν
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Re < 2,000 → Laminar
2,000–4,000 → Transitional
Re > 4,000 → Turbulent
ν(T) = 1.792×10⁻⁶ / (1 + 0.0337T + 0.000221T²) m²/s

1 m water = 0.0980665 bar = 9.80665 kPa = 1.42233 psi = 3.28084 ft
1 bar = 10.1972 m  |  1 psi = 0.70307 m  |  1 ft = 0.3048 m
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Head ↔ pressure assumes water at ρ = 1000 kg/m³ (≈4 °C).

hf = f × (L/D) × V²/(2g)  [Darcy-Weisbach]
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Re < 2,300 → Laminar: f = 64/Re
Re > 4,000 → Turbulent: Swamee-Jain explicit
f ≈ 0.25 / [log(ε/3.7D + 5.74/Re⁰·⁹)]²  (±3% of Colebrook-White)

F = P × A × 2sin(θ/2)  [resultant, N]
A = π D² / 4  [pipe bore area]
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90° bend: F = P × A × √2    Dead end: F = P × A
Block bearing area = F / σ_soil
Min embedment: 0.6 m (below frost / road loads)

1/√f = −2·log₁₀( (ε/D)/3.7 + 2.51/(Re·√f) )  (implicit — iterated)
Laminar (Re < 2300): f = 64 / Re
Swamee-Jain (explicit): f = 0.25 / [ log₁₀( ε/D/3.7 + 5.74/Re^0.9 ) ]²
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Feed f into Darcy-Weisbach: h_f = f · (L/D) · V²/2g

P_gross = 2 × SMYS × t / OD  [MPa]
P_allow = P_gross × DF / SF  [MPa]
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SMYS = Specified Min. Yield Strength (MPa)
t = wall thickness (mm)  |  OD = outside diameter (mm)
DF = design/location factor (0.40–0.72)
SF = safety factor (typically 1.25–1.5)

Δy/D = D_L · K_x · (W_c + W_L) / (0.149·PS + 0.061·E′)
W_c = γ · H  (soil prism pressure)
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PS & E′ in consistent stiffness units (kPa). Typical limits:
5% (steel/DI lined) · 7.5% long-term (GRP) · check product standard.

Uplift F_b = γ_w · (π/4 · D_o²)  [kN/m] (γ_w = 9.81 kN/m³)
Resisting = W_pipe + (γ_sat − γ_w)·D_o·H (submerged soil prism)
FoS = Resisting / Uplift  →  target ≥ 1.2
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Conservative: ignores backfill shear & pipe contents. Empty pipe = worst case.

Free movement: ΔL = α · L · ΔT
Fully restrained stress: σ = E · α · ΔT (independent of length)
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Free ΔL sizes expansion joints / loops; restrained σ sizes anchors.
Plastics (HDPE) move a lot but develop low stress (low E).

ΔP = ρ × a × ΔV  [Pa]
ΔP [bar] = ρ × a × ΔV / 100,000
Surge head [m] = ΔP / (ρ × g)
Critical closure time Tc = 2L / a
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ρ = 1,000 kg/m³  |  g = 9.81 m/s²
For instantaneous closure: ΔV = full flow velocity

a = √(K/ρ) / √(1 + K·D/(E·e))
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K = 2.1 × 10⁹ Pa  (bulk modulus, water)
ρ = 1,000 kg/m³
E = pipe Young's modulus (Pa)
D = internal diameter (m)
e = wall thickness (m)

Boyle's Law: P₀ × V_vessel = P_max × (V_vessel - V_water)
→ V_vessel = V_water × P_max_abs / (P_max_abs - P₀_abs)
P₀ = SF × P_min_abs − 1.013  [bar_g]
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All pressures converted to absolute (bar_a = bar_g + 1.013)
SF = 0.9 recommended (typical industry practice)
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⚠ V_water here is a rough proxy (½ · Q · 2L/a). True swing depends on the
column deceleration & pump inertia — use this only for first-pass sizing,
then confirm the vessel with a transient (water-hammer) simulation.

Critical (pipe) time: T_c = 2L / a
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T < T_c → rapid closure: full Joukowsky surge develops (ΔH = aV/g)
T ≥ T_c → slow closure: surge reduced (use Michaud ΔH = 2LV/gT)

Joukowsky (T ≤ T_c): ΔH = a·V₀ / g  (upper bound)
Michaud (T > T_c): ΔH = 2·L·V₀ / (g·T)
T_c = 2L/a  |  g = 9.81 m/s²
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Linear-closure estimate; real valve laws & line packing need a transient model.

Air rate ≈ water rate: Q_air = A_pipe · V (1:1 displacement)
Orifice: Q_air = C_d · A_o · √(2·Δp / ρ_air) → solve A_o → d_o
ρ_air ≈ 1.2 kg/m³ | Δp in Pa (subsonic screening)
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Vacuum (draining/burst) needs LARGE orifice; release needs small.
Sizing screen only — confirm with manufacturer flow curves (AWWA C512).

Total = P_working + ΔP_surge  |  Allowable = PN × factor
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Many codes permit a short-term surge allowance (factor 1.2–1.4 × PN).
Keep factor = 1.0 for a conservative steady + transient check.

Q = (1/n) × A × R^(2/3) × S^(1/2)  [m³/s]
A = π D² / 4  |  R = D/4 (full bore)
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Self-cleansing: V ≥ 0.6 m/s for sewers
Erosion limit: V ≤ 3.0 m/s for concrete

Q = (1/n) × A × R^(2/3) × S^(1/2)
Rectangular: A = b·y  |  P = b + 2y
Trapezoidal: A = (b + z·y)·y  |  P = b + 2y√(1+z²)
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Froude Fr = V / √(g·A/T)  |  Fr < 1 subcritical · Fr > 1 supercritical

Fr = V / √(g·y)  |  V = Q / (b·y)
Critical depth (rect): y_c = (q² / g)^(1/3), q = Q/b
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Fr < 1 subcritical (tranquil) · Fr = 1 critical · Fr > 1 supercritical (rapid)
Critical/near-critical flow is unstable — avoid designing channels there.

Avg day = pop × per-capita / 1000 [m³/d], then ÷ (1 − NRW)
Max day = Avg × MDF  |  Peak hour = Avg × PHF
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Treatment sized on max-day; network & pumps on peak-hour.
Typical MDF 1.4–1.8 · PHF 2.0–3.0 (smaller for larger populations).

Balancing = % × MDD  (diurnal equalization, typ. 20–35%)
Emergency = hours × average hourly demand
Fire reserve = fire flow × duration
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Total = balancing + emergency + fire. Confirm balancing % from a
mass-curve (Rippl) of the actual diurnal demand pattern.

V_total = V_operational + V_emergency + V_fire
V_operational = daily demand × operational % / 100
V_emergency = Q_max × duration (h)
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Operational: typically 15–25% of daily demand (EN 805)
Emergency: 8–24 h at max demand (project specific)
Fire reserve: per NFPA 13 / local code

Dose rate [kg/h] = Dose [mg/L] × Q [m³/h] / 1,000
Residual C = Applied dose − Chlorine demand
CT achieved = C × T_contact × (T₁₀/T)
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T₁₀/T = hydraulic efficiency (baffled tank ~0.7–0.9)
CT values from WHO 2011 / USEPA Guidance Manual

HRT = V / Q  |  Required V = Q × target HRT
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Typical: rapid mix 1–3 min · flocculation 20–40 min
sedimentation 2–4 h · chlorine contact ≥ 30 min (then check CT)

SOR = Q / A_surface  (m³/m²·d = m/d) ; A = π/4·D²
Weir loading = Q / weir length (m³/m·d)
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Typical: SOR 24–48 m/d (settling) · weir loading ≤ 250–375 m³/m·d

Mass (kg/day) = Dose(mg/L) × Q(m³/day) / 1000
Solution (L/day) = Mass / (strength fraction × density)
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Typical alum dose 10–60 mg/L; jar-test to confirm for your raw water.

V-notch 90°: Q = 1.38 · H^2.5
Rectangular (suppressed): Q = 1.84 · b · H^1.5 (Rehbock)
Cipolletti: Q = 1.86 · b · H^1.5  |  Q in m³/s, H & b in m
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Valid for free-flow, fully ventilated nappe; measure H ≥ 3–4× upstream of crest.

Minimize: TC/m = CRF × k × D + 8ρfQ³ × C_e × H / (π²η × D⁵ × 1000)
Setting d(TC)/dD = 0 → D_econ = [40ρfQ³C_eH / (π²η × CRF × k × 1000)]^(1/6)
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CRF = i(1+i)^N / [(1+i)^N − 1] (Capital Recovery Factor)
k = pipe cost coefficient [$/m per mm of diameter]
Q in m³/s · ρ = 1000 kg/m³ · C_e in $/kWh
Source: Optimum Pipeline Design (Walski et al.)