$($iii$)$ Let us say there are two non$-$dimensional numbers $M$ and $N.$ Then prove that the

solution of the governing differential equation can be written as $\frac{T-T_{}}{T_i-T_{}}=e^{-(MN)},$ Identify

these two non$-$dimensional numbers $M($independent of time$)$ and $N($dependent of time$)$

and write down their expressions.

$($iv$)$ The temperature of a hot fluid stream having a convective heat, transfer coefficient

of $485W{m}^{-2}{K}^{-1}$ is to be measured by a thermocouple. Let us say that the thermocou$-$

ple bead may be assumed to be a perfect sphere with diameter $1$ mm $.$ The thermocouple

initially at room temperature of $20C$ is placed into the hot stream of $100C.$ The thermo$-$

physical propertis are: $=8900kg{m}^{-3},k=400W{m}^{-1}{K}^{-1},c_p=385.Jk{g}^{-1}{K}^{-1}.$

Calculate the non$-$dimensional number $M.$ Calculate the other non$-$dimensional num$-$

ber $N$ for the thermocouple temperature reading to be $92C.$