Measuring Temperature — Question 7

Question 7

Stem conduction biases an immersed probe

A probe exchanges energy with a bath through $G_b=.90\,\mathrm{W/K}$ and with ambient air at $290\,\mathrm K$ through its stem with $G_a=.10\,\mathrm{W/K}$. Its steady reading is $317\,\mathrm K$. The model is $G_b(T_b-T_s)+G_a(T_a-T_s)=0$.

1. Infer the bath temperature and the reading bias.

2. At $T_b=320\,\mathrm K$, find the largest ratio $G_a/G_b$ that keeps the magnitude of the bias below or equal to $.20\,\mathrm K$.

Question 7 - Solution

Stem conduction biases an immersed probe

(a) Method and calculation

$$T_b=T_s+\frac{G_a}{G_b}(T_s-T_a)=317+\frac{.10}{.90}(27)=320\,\mathrm K.$$$$T_s-T_b=-3\,\mathrm K.$$The probe is cooled through the stem despite reaching a steady reading.

(b) Interpretation and further calculation

For $r=G_a/G_b$,$$T_b-T_s=\frac{r}{1+r}(T_b-T_a)=\frac{30r}{1+r}\,\mathrm K.$$$$\frac{30r}{1+r}\le.20\ \Longrightarrow\ r\le\frac1{149}\approx.006711.$$Deeper immersion or a less conducting stem can change this ratio.

Physical check

This systematic error survives long waiting; unlike transient lag it is maintained by a second thermal path.