= Thermal loading by a contact temperature sensor
\b[Thermal loading by a contact temperature sensor] is the disturbance of an object's <temperature> caused by exchanging <heat> with the <temperature sensor> used to measure it. A sensor initially colder than a small object draws <heat> from the object: the final reading can be accurate for the disturbed system while differing from the object's original <temperature>.
An illuminating idealization treats the object and sensor as two uniform bodies with constant <heat capacities> $C_o$ and $C_s$, joined by <thermal conductance> $G$ and isolated from everything else. Their <energy conservation> equations are
$$
C_o\dot T_o=-G(T_o-T_s),\qquad C_s\dot T_s=G(T_o-T_s).
$$
Adding the equations shows that $C_oT_o+C_sT_s$ is constant. Consequently,
$$
T_\infty=\frac{C_oT_{o0}+C_sT_{s0}}{C_o+C_s},\qquad
T_{o0}-T_\infty=\frac{C_s}{C_o+C_s}(T_{o0}-T_{s0}).
$$
For $C_o=0.20\,\mathrm{J/K}$, $C_s=0.020\,\mathrm{J/K}$, $T_{o0}=80\,{}^{\circ}\mathrm{C}$ and $T_{s0}=20\,{}^{\circ}\mathrm{C}$, the common equilibrium is $74.55\,{}^{\circ}\mathrm{C}$. The measurement has cooled the object by about $5.45\,\mathrm{K}$.
Subtracting the two differential equations gives an exponentially decaying <temperature> difference with <time constant>
$$
\tau_{\mathrm{difference}}=\frac{C_oC_s}{G(C_o+C_s)}.
$$
Increasing the contact <thermal conductance> makes the disturbance happen faster without changing this isolated model's final equilibrium. Reducing the sensor <heat capacity> reduces the disturbance. A real object also exchanges <heat> with its surroundings, while sensor leads can provide another <heat transfer> path.
An electrically read sensor can additionally generate <Joule heating>. That is a separate loading mechanism: the thermometry guide measures resistance-thermometer self-heating by changing the measuring <electric current> and extrapolating toward zero dissipated <electric power>.<Guide on Secondary Thermometry: Industrial Platinum Resistance Thermometers by Jonathan Pearce and others>[§4.1, printed pp. 24–26, two-current self-heating evaluation]{ref}
Back to article page
