Math & Numbers

Compute the effectiveness (ε) of a heat exchanger by the ε-NTU method. C*=C_min/C_max (0..1), NTU=U·A/C_min, q_max=C_min·(T_h,in-T_c,in), and ε=q_actual/q_max. Supports four arrangements: Parallel flow ε=[1-exp(-NTU(1+C*))]/(1+C*); Counter flow ε=[1-exp(-NTU(1-C*))]/[1-C*·exp(-NTU(1-C*))], or NTU/(1+NTU) when C*=1; Shell-and-tube 1-2 ε=2/[1+C*+√(1+C*²)·(1+exp(-NTU√(1+C*²)))/(1-exp(-NTU√(1+C*²)))]; Crossflow (both unmixed) ε=1-exp{(NTU^0.22/C*)·[exp(-C*·NTU^0.78)-1]}. When C*=0 (phase change on one side, boiler/condenser) every arrangement gives ε=1-exp(-NTU). Optionally supply T_h,in and T_c,in to recover the actual heat-transfer rate q=ε·q_max and both outlet temperatures. Temperatures are used only as differences: Δ°C=ΔK and Δ°F×5/9=ΔK.

Math & Numbers

Estimate the heating (heat-loss) load of a room by the simplified steady-state method. Envelope transmission loss Q_trans = Σ(A_i·U_i·ΔT) over surfaces entered one per line as 'area,U' (U in W/(m²·K)). Cold-air infiltration loss Q_inf = 0.018·ACH·V·ΔT (W), where 0.018 W·h/(m³·K) ≈ ρ·c_p/3600. Total load = Q_trans + Q_inf, optionally multiplied by a safety factor (default 1.0). ΔT = indoor - outdoor (heating, >0). Temperature in °C/K/°F (only differences matter), area in m²/ft².

Math & Numbers

Compute 1-D steady-state heat conduction through a flat slab (Fourier's law): heat flux q = k·ΔT/d (W/m²), heat flow rate Q = k·A·ΔT/d (W), and thermal resistance R = d/(k·A) (K/W). k is the thermal conductivity (W/(m·K)); ΔT is the temperature difference (K; a °C difference equals a K difference, a °F difference is converted by ×5/9); d the slab thickness; A the cross-section area. ΔT may be negative (indicating reverse heat flow), but k, d and A must be positive. Thickness in m/cm/mm, area in m²/cm².

Math & Numbers

Compute convective heat transfer (Newton's law of cooling): heat flux q = h·ΔT (W/m²), heat flow rate Q = h·A·ΔT (W), and convective thermal resistance R_conv = 1/(h·A) (K/W). h is the convective heat-transfer coefficient (W/(m²·K)); ΔT is the temperature difference between the surface and the fluid (K; a °C difference equals a K difference, a °F difference is converted by ×5/9); A the heat-transfer area. ΔT may be negative (indicating reverse heat flow), but h and A must be positive. Area in m²/cm².

Math & Numbers

Compute the Log Mean Temperature Difference (LMTD) of a heat exchanger for parallel or counter flow. Parallel flow: ΔT₁ = T_h,in - T_c,in and ΔT₂ = T_h,out - T_c,out; counter flow: ΔT₁ = T_h,in - T_c,out and ΔT₂ = T_h,out - T_c,in. LMTD = (ΔT₁ - ΔT₂)/ln(ΔT₁/ΔT₂), or ΔT₁ when ΔT₁ = ΔT₂. A non-positive terminal difference (temperature cross) is physically impossible and is rejected. Optionally, with the overall heat transfer coefficient U (W/(m²·K)) and the heat transfer area A (m²), the heat transfer rate Q = U·A·LMTD (W) is returned. Temperatures are used only as differences: Δ°C = ΔK and Δ°F ×5/9 = ΔK; the LMTD is reported in K.

Math & Numbers

Compute thermal radiation from a blackbody/grey body (Stefan-Boltzmann law): blackbody emissive power E_b = σ·T⁴ (W/m²), total radiated power Q_rad = ε·σ·A·T⁴ (W); with an optional surrounding temperature T₀ it also computes the net radiative exchange Q_net = ε·σ·A·(T⁴-T₀⁴) (W). σ = 5.670374419e-8 W/(m²·K⁴); ε is the emissivity (0 < ε ≤ 1, blackbody ε=1); T is the ABSOLUTE temperature (K) — °C and °F are first converted to K (this is an absolute temperature, not a difference); A is the radiating area. Area in m²/cm².

Math & Numbers

Compute complex admittance Y = 1/Z = G + jB for series or parallel RLC circuits: G = R/|Z|², B = −X/|Z|². Returns conductance, susceptance, magnitude and phase angle. Supports any subset of R/L/C.

Math & Numbers

Compute complex impedance, magnitude |Z| and phase angle φ for series or parallel RLC circuits: X_L=2πfL, X_C=1/(2πfC), |Z|=√(R²+X²), φ=atan2(X,R). Supports any subset of R/L/C and an optional linear frequency sweep.

Math & Numbers

Calculate the Rankine active earth pressure on a smooth vertical wall backfilled with cohesionless soil. K_a = tan²(45°−φ/2). Optional uniform surcharge q. Returns the base pressure p₀, the total force P, and the resultant height.

Finance

Calculate the true Annual Percentage Rate (APR) of a loan when upfront fees are involved. The nominal interest rate sets the monthly payment, but fees reduce the net cash received, raising the effective APR. Uses Newton's method to solve the annuity present-value equation. Shows APR, nominal rate, monthly payment, and the APR premium.

Finance

Convert an amount between currencies using live exchange rates from a free, no-key API (open.er-api.com). Supports 160+ world currencies. Shows the converted amount, the exchange rate used, a comparison table against popular currencies, and the rate update timestamp.

Finance

Project investment growth with initial principal plus monthly contributions and compound interest. Supports compounding frequency, inflation-adjusted real value, and yearly contribution escalation, with a yearly breakdown table and nominal vs real balance curves.