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Consider two rigid bodies forming an isolated system, with fixed composition and constant whole-body heat capacity at constant volume values $C_1=1\mathrm{kJ}{\mathrm{K}}^{-1}$ and $C_2=3\mathrm{kJ}{\mathrm{K}}^{-1}$. Their initial temperatures are $T_{1,0}=300\mathrm{K}$ and $T_{2,0}=400\mathrm{K}$. Assume each body remains internally uniform, their contact stores negligible energy, no phase transition occurs and no work is done or extracted. These are stipulated model values, not material measurements.

Because $d{U}_i=C_{i}d{T}_i$, write $U_i=a_i+C_i{T}_i$ with arbitrary fixed reference constants $a_i$. The conserved relative internal energy isThis is an energy reference convention, not a claim about actual heat capacities down to zero temperature. At thermal equilibrium the two temperatures equal $T_f$, so the first law of thermodynamics givesThe larger heat capacity pulls the common temperature closer to its initial value: body 1 warms by $75\mathrm{K}$, while body 2 cools by $25\mathrm{K}$.

To predict a timescale, stipulate a constant contact conductance $G=1/R_{\mathrm{th}}=0.1\mathrm{kJ}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$, the reciprocal of the contact's thermal resistance. The rate of heat transfer into body 1 is $G(T_2-T_1)$, givingAdding proves conservation of $E$. Subtracting gives exponential decay of the temperature difference, with time constantand thereforeExact equality is approached as $t\to \infty$ in this ideal model. Changing $G$ changes the rate but not $T_f$: equilibrium and the speed of reaching it are separate predictions. Real bodies can have internal temperature gradients, changing heat capacities and contact losses, so the plotted $7.5\mathrm{s}$ is an assumed model timescale.

Thermocouple voltage calibration relates an actual assembled sensor's electromotive force to independently established junction temperatures. It must be distinguished from selecting a nominal letter-type conversion curve. In Thermocouples Calibrations Services by National Institute of Standards and Technology (opening method description and bath-calibration table), NIST describes comparison with a maintained reference thermocouple, comparison with a standard platinum resistance thermometer in controlled heat baths, and fixed-point calibration at the freezing points of zinc, aluminum, silver and gold on the International Temperature Scale of 1990. These are independent thermal references, unlike a test that checks only a voltage-input circuit.^[ref]

For a concrete electrical check, the SCC-TC guide describes shorting the input after 30 minutes of powered operation and adjusting its offset to a zero-voltage readout. Its separate calibrator procedure requires equal connection temperatures at the calibrator and module, measures the supplied voltage, converts it to temperature, and compares with the calibrator setting. Such an injected-signal test checks the readout chain; it does not expose a real measuring junction to an independently known bath temperature.^[ref]

Useful additional checks follow from the model: log the raw voltage, the reference temperature, and the final inferred temperature separately; hold $T_h$ stable while changing $T_c$ and test whether cold-junction compensation preserves the inferred $T_h$; compare against an independent thermometer at several stable heat bath settings. Check immersion and repeatability instead of assuming a steady display establishes thermal equilibrium. These are proposed diagnostic checks, not new experimental results. The equal-temperature and polarity controls are already described under Seebeck effect.

Limits arise separately from the thermocouple reference function, sensor-wire changes, thermal contact, terminal temperature differences, and the voltage-input circuit. In particular, the SCC-TC guide warns of connector-to-module temperature gradients and voltage drops caused by the fault-detection electric current flowing through lead electrical resistance.^[ref] A fitted curve or compensation algorithm cannot make all these errors vanish. No single accuracy is assigned to every thermocouple.

A thermoelectric effect couples heat transfer and the transport of electric charge: a temperature gradient can generate an electromotive force, while an electric current can carry heat and cause heating or cooling at a junction of different materials. This coupling makes electrical measurement of temperature possible through a thermocouple, and permits conversion of a sustained flow of heat into electrical work. A finite open-circuit voltage does not itself deliver power: electrical power requires an electric current through a load. Thermoelectric conversion therefore involves both electromagnetism and thermodynamics. ^[ref].

The combination $S^2\sigma$ of Seebeck coefficient and electrical conductivity. Increasing it alone need not increase ZT if thermal conductivity rises as well.

The Thomson effect is distributed reversible heating or cooling when an electric current flows through a homogeneous material with a temperature gradient. It is distinct from junction Peltier effect heating and even-in-current Joule heating.

In the tropical free troposphere, small Coriolis parameter and rapid internal gravity wave adjustment keep horizontal temperature gradients weak. An idealized model therefore suppresses horizontal temperature perturbations at leading order and balances diabatic heating by vertical advection of the reference stratification. This is not a claim that surface temperature or moisture is spatially uniform, nor is it ordinary midlatitude geostrophic balance.