Question 7.8.5: A thermocouple can be used to measure temperature. The elect......
A thermocouple can be used to measure temperature. The electrical resistance of the device is a function of the temperature of the surrounding fluid. By calibrating the thermocouple and measuring its resistance, we can determine the temperature. Because the thermocouple has mass, it has thermal capacitance, and thus its temperature change (and electrical resistance change) will lag behind any change in the fluid temperature.
Estimate the response time of a thermocouple suddenly immersed in a fluid. Model the device as a sphere of copper constantin alloy, whose diameter is 2 mm, and whose properties are ρ = 8920 kg/m³, k = 19 W/m·°C, and c_{p} = 362 J/kg·°C. Take the convection coefficient to be h = 200 W/m²·°C.
First compute the Biot number N_{B} to see if a lumped-parameter model is sufficient. For a sphere of radius r,
L={\frac{V}{A}}={\frac{(4/3)\pi r^{3}}{4\pi r^{2}}}={\frac{r}{3}}={\frac{0.001}{3}}=3.33\times10^{-4}The Biot number is
N_{B}={\frac{h L}{k}}={\frac{200(3.33\times10^{-4})}{19}}=0.0035which is much less than 0.1. So we can use a lumped-parameter model.
Applying conservation of heat energy to the sphere, we obtain the model:
c_{\rho}\rho V{\frac{d T}{d t}}=h A(T_{o}-T)where T is the temperature of the sphere and {{T}}_{o} is the fluid temperature. The time constant of this model is
\tau=\frac{c_{p}\rho}{h}\frac{V}{A}=\frac{362(8920)}{200}3.33\times10^{-4}=5.38\,\mathrm{s}The thermocouple temperature will reach 98% of the fluid temperature within 4τ = 21.5 s.