Question

A small body (Biot number < 0.1) has temperature variation as $T=A+B{e}^{C\tau },\tau =$ time in seconds
Where A, B, and C are constants.
If that small body is used as thermocouple its time constant is given by

• A. -1/C
• B. 1/C
• C. -C
• D. C

Answer 1

The correct option is A -1/C

In luped analysis temperature variation is given by
$\frac{T-T_{\infty }}{T_i-T_{\infty }}=e^{\frac{hA}{\rho vc}\tau }$ ...(i)
Given equation, $T=A+B{e}^{C\tau }$
$\frac{T-A}{B}=e^{C\tau }$ ...(ii)
Compare (i) and (ii)
$A=T_{\infty },B=T_i-T_{\infty }$
$C+\frac{-hA}{\rho vc}$
Now time constant is given by
${\tau }^0=\frac{\rho vc}{hA}$
${\tau }^0=\frac{-1}{C}$