Math & Numbers

Jacobi/Legendre symbol with deterministic Miller–Rabin primality, Tonelli–Shanks square roots for prime moduli, and brute-force resolution for small composite moduli.

Math & Numbers

Steady-state M/M/1 metrics: ρ, P0, L, Lq, W, Wq, state probabilities, and Little's-law cross-checks; unstable λ ≥ μ inputs are rejected.

Math & Numbers

Steady-state M/M/c metrics with the Erlang-C formula: offered load, P0, Pw, Lq, L, W, Wq plus Little's-law cross-checks; unstable λ ≥ cμ inputs are rejected.

Math & Numbers

Convert pH ↔ [H⁺] in mol/L, mmol/L, µmol/L or nmol/L with the full unit table — pH 7.40 ↔ 39.8 nmol/L.

Math & Numbers

Calculate Poisson's ratio ν = −ε_transverse / ε_longitudinal and solve any of the three unknowns: ratio, lateral strain, or longitudinal strain. Tensile ε_long > 0 produces lateral contraction ε_lat < 0. Typical values: steel ≈ 0.30, aluminum ≈ 0.33, rubber ≈ 0.49, cork ≈ 0. Theoretical isotropic range 0 ≤ ν ≤ 0.5.

Math & Numbers

Compute the full set of moist-air (psychrometric) state properties from dry-bulb temperature plus one humidity input (relative humidity φ, wet-bulb temperature, dew-point temperature, or humidity ratio W). Returns humidity ratio W (kg/kg and g/kg), relative humidity φ (%), dew-point, wet-bulb, enthalpy h (kJ/kg dry air), specific volume v (m³/kg dry air), partial vapour pressure P_w, and saturation pressure P_ws. Uses the Magnus saturation-pressure fit; wet-bulb is solved iteratively from the thermodynamic psychrometric equation. Default standard atmosphere 101.325 kPa (adjustable). Temperature in °C/K/°F; all reported temperatures are in °C.

Math & Numbers

Compute the flow rate through a thin-plate orifice for an incompressible fluid (ISO 5167 Bernoulli form with discharge coefficient C_d). Orifice area A=π·d²/4; volumetric flow Q=C_d·A·√(2·ΔP/ρ); mass flow ṁ=C_d·A·√(2·ρ·ΔP); throat velocity v_orifice=C_d·√(2·ΔP/ρ). If the upstream pipe diameter D is supplied the diameter ratio β=d/D is also returned. ΔP≥0, ρ>0, C_d in 0..1 (default 0.61 for a sharp-edged plate). Diameter in m/cm/mm, pressure in Pa/kPa/bar/atm/psi, density in kg/m³/g/cm³; volumetric flow reported in m³/s, L/s, L/min and m³/h.

Math & Numbers

Compute the overall heat transfer coefficient for series thermal resistances (flat-wall model): total resistance R_total = Σ R_i (K/W), thermal conductance G = 1/R_total (W/K), area-based coefficient U = 1/(R_total·A) (W/(m²·K)), and if a temperature difference is given the heat flow rate Q = ΔT/R_total (W). Each R_i is a layer resistance already containing the area factor (e.g. convection R_conv=1/(h·A), conduction R_cond=d/(k·A), fouling R_foul=R_f''/A). Enter one resistance (K/W) per line, at least one, all positive. ΔT may be negative (reverse heat flow); a °C difference equals a K difference, a °F difference is converted by ×5/9.

Math & Numbers

Apply the pump affinity laws to the same pump at different speeds or with a different impeller diameter. Speed mode: ratio = n2/n1; diameter mode: ratio = D2/D1. Then Q2 = Q1·ratio, H2 = H1·ratio², P2 = P1·ratio³. Q1, H1 and P1 are optional baseline-condition inputs; if all three are omitted, only the ratio is returned. Flow in m³/s/L/s/L/min/m³/h, head in m/ft, power in W/kW/hp, impeller diameter in m/cm/mm; SI equivalents (m³/s, m, W) are also reported.

Math & Numbers

Compute the total dynamic head H = (p_d - p_s)/(ρ·g) + Δz + (v_d² - v_s²)/(2g) of a centrifugal pump (m), then the fluid power P_fluid = ρ·g·Q·H and shaft power P_shaft = P_fluid/η. p_d and p_s are gauge pressures (suction may be negative for suction lift). Nozzle velocities v = 4Q/(π·D²) come from the flow rate and the discharge/suction nozzle diameters; if a nozzle diameter is left blank its velocity head is omitted. Pressure in Pa/kPa/bar/atm/psi, flow in m³/s/L/s/L/min/m³/h, density in kg/m³/g/cm³; power output in W/kW/hp.

Math & Numbers

Compute the Darcy friction factor: laminar closed-form f=64/Re for Re<2000, or iterative Colebrook-White solution for transitional/turbulent flow, with the explicit Haaland approximation always returned as a cross-check. Reports relative roughness ε/D, absolute roughness (m), diameter (m), flow regime (laminar/transitional/turbulent), effective method and iteration count. Roughness ε in m/mm/µm, diameter D in m/cm/mm.

Math & Numbers

Compute the Darcy-Weisbach major (friction) pressure drop in a straight pipe: ΔP = f·(L/D)·(ρ·v²/2) Pa and head loss h_f = f·(L/D)·v²/(2g) m. The user supplies the Darcy friction factor f (not the Fanning factor). Length in m/km/ft, diameter in m/cm/mm/inch, density in kg/m³/g/cm³/lb/ft³ — all normalised to SI internally. Returns ΔP in Pa, kPa and bar, and head loss in m and ft. Gravity defaults to 9.81 m/s² and may be overridden.