Health

Design the dosing interval τ from the steady-state peak/trough concentration ratio. For one-compartment linear kinetics at steady state, the peak-to-trough ratio over one interval is P = Cmax,ss / Cmin,ss = e^(kτ) where k = ln2 / t½. Mode A: given a target peak/trough ratio, solve for the dosing interval τ = ln(P) · t½ / ln2. Mode B: given a chosen interval τ, compute the resulting peak/trough ratio P = e^(kτ). Note: P reflects fluctuation WITHIN one interval and is NOT the accumulation index (R = 1/(1-e^(-kτ))). Valid only for linear one-compartment kinetics, steady state, fixed dose and interval. Not medical advice.

Health

Calculate the drug accumulation index R under repeated fixed-interval dosing with linear first-order elimination: R = 1 / (1 - e^(-kτ)), where k = ln2 / t½ and τ is the dosing interval. R is the ratio of the steady-state peak concentration to the peak concentration after the first dose — it quantifies how much the body burden rises at steady state versus a single dose. For example, dosing once per half-life (τ = t½) gives R = 2. The term e^(-kτ) is the fraction of peak concentration remaining at the end of one interval. Valid only for linear first-order elimination with fixed dose and interval — NOT for Michaelis-Menten drugs (phenytoin) or non-linear kinetics. Not medical advice.

Health

Convert an elimination half-life to the first-order elimination rate constant: k = ln2 / t½ = 0.693 / t½. Also reports the fraction of drug remaining after 1/2/3/4/5 half-lives (50%, 25%, 12.5%, 6.25%, 3.125%), the number of half-lives required to reach steady state (≈4 for 94%, ≈5 for 97%), and the time to ~99% elimination (≈6.64 half-lives). Valid only for first-order (linear) elimination — NOT for zero-order or Michaelis-Menten kinetics (phenytoin, high-dose ethanol, high-dose aspirin). Not medical advice.

Health

Calculate drug clearance CL in two modes. (A) Definition: CL = elimination rate / concentration, valid for any pharmacokinetic model. (B) Linear first-order: CL = k · Vd (or derive k from t½ via k = ln2 / t½). Also reports the relationship t½ = 0.693 · Vd / CL. CL is a definitional parameter independent of Vd. Linear mode applies only to first-order elimination — NOT to Michaelis-Menten drugs (phenytoin, high-dose ethanol). Reference: theophylline CL ≈ 2.8 L/h in a 70 kg non-smoking adult. Not medical advice.

Health

Estimate plasma drug concentration at a chosen time after an IV bolus using a one-compartment model with first-order elimination: C(t) = (Dose / Vd) · e^(-k·t), where k = ln2 / t½. Includes initial concentration C0, elimination rate constant k, percent remaining, and number of half-lives elapsed. Valid only for linear pharmacokinetics — NOT for Michaelis-Menten drugs (phenytoin, high-dose ethanol) or multi-compartment distribution. Reference: theophylline Vd ≈ 0.45 L/kg, t½ ≈ 8 h. Not medical advice.

Math & Numbers

Convert dynamic (absolute) viscosity between Pa·s (pascal-second, SI base, = N·s/m² = kg/(m·s)), cP (centipoise, 1 cP = 0.001 Pa·s), and lbf·s/ft² (imperial, 1 = 47.88025898 Pa·s). Converts via Pa·s to the target unit and also lists the equivalent value in all three units for reference. Reference values: water 20°C ≈ 1.002 cP, glycerol 20°C ≈ 1412 cP, air 20°C ≈ 0.0181 cP.

Math & Numbers

Compute the air dew-point temperature from dry-bulb temperature and a humidity reading using the Magnus saturation-pressure equation: P_ws(T) = 0.6108·exp(17.27·T/(T+237.3)), P_w = φ·P_ws, T_dp = 237.3·ln(P_w/0.6108) / (17.27 − ln(P_w/0.6108)). Also returns the saturation vapour pressure P_ws, actual partial pressure P_w, and absolute humidity ρ_v (kg/m³). Three input modes: relative humidity φ (natural ventilated), August natural-ventilated wet-bulb T_w, or aspirated (Assmann) wet-bulb T_wa. Temperature in °C/K/°F; all reported temperatures are in the selected Temperature Unit.

Math & Numbers

Compute the drag force on a body in a fluid stream: F_D = ½·ρ·v²·C_D·A (N), where ρ is the fluid density, v the free-stream velocity, C_D the drag coefficient (shape- and Reynolds-dependent), and A the reference (frontal projected) area. A built-in shape library supplies typical high-Re C_D values: sphere 0.47, hemisphere 0.42, long cylinder 0.81, disk/flat plate 1.17, cube 1.05, streamlined airfoil 0.04, long streamlined ellipsoid 0.07, cone 0.50; or choose 'Custom' to enter C_D directly. Optionally, with dynamic viscosity μ and characteristic length L, the Reynolds number Re = ρ·v·L/μ is computed, and the power dissipated by drag P = F_D·v (W). Density in kg/m³/g/cm³, velocity in m/s/km/h, area in m²/cm²/mm², length in m/cm/mm.

Math & Numbers

Size a duct cross-section from the air flow rate Q and the design mean velocity v: area A = Q/v. Circular duct diameter D = √(4A/π). Rectangular duct with aspect ratio r = a/b gives b = √(A/r) and a = r·b, plus the ASHRAE equivalent diameter D_eq = 1.30·(a·b)^0.625/(a+b)^0.25. Flow rate in m³/s/m³/h/CFM, velocity in m/s; dimensions reported in mm and inches.

Math & Numbers

Convert between the three common refrigeration efficiency metrics: COP (dimensionless, W/W), EER (Btu/(W·h)), and SEER (seasonal Btu/(W·h)). Standard relations: COP = EER / 3.412, EER = COP × 3.412, SEER = EER / α (α seasonal weighting factor, default 0.875, adjustable 0.8–0.91 to match regional standards such as ARI 210/240). Enter any one metric and all three are reported.

Data Processing

Evaluate bounded Excel-style formulas against JSON cell values and explain results, references, and spreadsheet errors.

Math & Numbers

Look up and compute the maximum deflection for four classic beam cases: cantilever with end point load (δ=PL³/3EI), cantilever with UDL (δ=qL⁴/8EI), simply supported with mid-span point load (δ=PL³/48EI), and simply supported with UDL (δ=5qL⁴/384EI).