Question

3 Lumpy Junction A thermocouple junction, which may be approximated as a sphere, is to be used for temperature measurement in a gas stream. The convection coefficient between the junction surface and the gas is h = 200 W m-2 K-1. The junction properties are k = 35 W m-1 K-1, c = 500 J kg-1 K-1, and p = 8000 kg m-3. Determine the junction diameter needed for the thermocouple to have a time constant of t = 0.8 s. If the junction is at 20°C and is placed in a gas stream at 250 °C, how long will it take for the junction to reach 249 °C? [4.45]

Answer 1

# time constant is given as

$\tau =\frac{\rho Vc}{hA}$ .............(1)

$\frac{V}{A}=\frac{\frac{4\pi r^{3}}{3}}{4\pi r^{2}}=\frac{r}{3}=\frac{d}{6}$

from equation (1)

$0.8=\frac{8000\times 500\times d}{200\times 6}\Rightarrow d=0.24mm$

# time required to reach 249 ^oC

T - TO T; - TO

$\Rightarrow t=-\tau ln\left | \frac{T-T_{0}}{T_{i}-T_{0}} \right |=-0.8ln\left | \frac{249-250}{20-250} \right |=4.35sec$