Q$.3$ A mercury thermometer is registering room temperature of $20C.$ Suddenly it is placed in a hot oil bath

maintained at $80C.$ The dynamic response of mercury thermometer is shown in Table $2.$

a$)$ Derive the transfer function of mercury thermometer highlighting the significance of assumptions.

b$)$ Estimate the time constant of mercury thermometer using the following three methods:

$($i$)$ Intial slope method, $($ii$)63\mathrm{.}2\%$ method, $($iii$)$ Linear regression method.

c$)$ Derive the formula used for each method and comment on the accuracy.