Temperature formulas

A change of 1 K equals a change of 1 °C and 1.8 °F. Use 273 when the question does.

T in °C, a and b depend on the pair. dE/dT is the thermoelectric sensitivity (Seebeck coefficient of the pair). For hot junction T_1 and cold junction T_2: E = E(T_1) - E(T_2).

For E = aT + bT^2 with a > 0 and b < 0. The emf is maximum when the hot junction is at T_n (where dE/dT = 0), whatever the cold-junction temperature T_c. It returns to zero at the inversion temperature T_i, which lies as far above T_n as T_c lies below it. With the cold junction at 0 °C, T_i = -a/b = 2T_n.

N identical thermocouples in series, with all hot junctions at the hot temperature and all cold junctions at the cold one. Output and sensitivity are N times those of one thermocouple. It is used to detect radiant power.

T in °C, α referred to 0 °C. Good over a modest range.

α_0 is referred to 0 °C and α_{T1} to T_1 (°C). Both describe the same linear law. For example, α_0 = 0.00385 /°C gives α_20 = 0.00385/1.077 = 0.003575 /°C.

For industrial platinum A = 3.9083 x 10^-3 /°C, B = -5.775 x 10^-7 /°C^2, C = -4.183 x 10^-12 /°C^4. Sensitivity falls slightly as T rises because B is negative.

R_L is the resistance of each lead. The lead resistance is assumed constant and the linear law is used.

Bridge with supply V_s, the RTD in one arm and the other three arms equal to R_0, so it balances at 0 °C. The approximation needs αT much less than 2. The exact output is slightly non-linear in T.

R_0 is the resistance at reference temperature T_0 (kelvin), β is the material constant in kelvin. R falls as T rises.

T_1, T_2 in kelvin, R_1 and R_2 the resistances at them.

Negative for NTC, and its magnitude falls as T rises. At room temperature it is a few per cent per kelvin.

T in kelvin, R in ohms. A, B and C are found from three calibration points. It fits a thermistor more closely over a wide range than the single-β law.

For two matched transistors at current-density ratio n. kT/q is about 25.85 mV at 300 K. The output is linear in absolute temperature.

T in kelvin. With surroundings at T_s, the net emission is εσ(T^4 - T_s^4).

T_a is the black-body (apparent) temperature the pyrometer shows, T is the true surface temperature in kelvin. Valid when radiation from the surroundings is neglected and ε is the total emissivity.

The wavelength at which a black body's spectral emission peaks falls as its temperature rises. T in kelvin.

T_a is the brightness temperature read, ε_λ the spectral emissivity at wavelength λ, temperatures in kelvin. Since ln ε_λ < 0, the true temperature is above the reading.

Ratio of spectral radiances at two wavelengths, under the Wien approximation. For a grey body (ε_1 = ε_2) the emissivity cancels, so the temperature follows from the ratio alone.

T_m is the sensor temperature, T_f the fluid temperature, m the mass, c the specific heat, h the film coefficient, A the surface area. A small, light sensor has a small τ.

The fluid steps from the initial temperature T_0 to T_f at t = 0. The sensor reaches 63.2 % of the step at t = τ, 90 % at 2.303τ and about 98 % at 4τ.

For a fluid temperature rising at a steady rate r (°C/s), after the transient dies away the reading lags the fluid by rτ in temperature, which is a time lag of τ.