Tate–Shafarevich group

Searle's bar method is an experimental procedure to measure thermal conductivity of material. A bar of material is being heated by steam on one side and the other side cooled down by water while the length of the bar is thermally insulated. Then the heat ΔQ propagating through the bar in a time interval of Δt is given by

${\left(\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}\right)}_{\mathrm{b}\mathrm{a}\mathrm{r}}=-kA\frac{\mathrm{\Delta }{T}_{\mathrm{b}\mathrm{a}\mathrm{r}}}{L}$

where

• ΔQ is the heat supplied to the bar in time Δt
• k is the coefficient of thermal conductivity of the bar.
• A is the cross-sectional area of the bar,
• ΔT_bar is the temperature difference of both ends of the bar
• 'L is the length of the bar

and the heat ΔQ absorbed by water in a time interval of Δt is:

${\left(\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}=C_{\mathrm{w}}\frac{\mathrm{\Delta }m}{\mathrm{\Delta }t}\mathrm{\Delta }{T}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$

where

• C_w is the specific heat of water,
• Δm is the mass of water collected during time Δt,
• ΔT_water is difference in the temperature of water before and after it has gone through the bar.

Assuming perfect insulation and no energy loss, then

${\left(\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}\right)}_{\mathrm{b}\mathrm{a}\mathrm{r}}={\left(\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$

which leads to

$k=-C_{\mathrm{w}}\frac{L}{A}\frac{\mathrm{\Delta }m}{\mathrm{\Delta }t}\frac{\mathrm{\Delta }{T}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}{\mathrm{\Delta }{T}_{\mathrm{b}\mathrm{a}\mathrm{r}}}$

References

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