First slide

Heat transfer-Radiation

Question

A body in a room cools from $85^{\circ} c$ to $80^{\circ} c$ in 5 minutes. The time taken to cool from $80^{\circ} c$ to $75^{\circ} c$ is

Moderate

A

5 minutes
B

<5 minutes
C

>5 minutes
D

either less or more than five minutes.

Solution

$\begin{array}{l} θ_{1} = 85^{\circ} {c θ}_{2} = 80^{\circ} c \\ θ_{2} = 80^{\circ} {c θ}_{3} = 75^{\circ} c \\ t_{1} =5 minutes \\ t_{2} = ? \\ θ_{0} = constant=temperature of surrounding \end{array}$
$\begin{array}{rcl} f r o m n e w t o n l a w o f c o o l i n g \\ \frac{dθ}{{dt}_{1}} =-k (\frac{θ_{1} {+θ}_{2}}{2} {-θ}_{0}) ---(1) \\ substitute given values in above equation \\ \frac{θ_{1} {-θ}_{2}}{5} & = & - k (\frac{85+80}{2} {-θ}_{0}) \\ \frac{85-80}{5} & = & - k (\frac{82.5}{2} {-θ}_{0}) \\ \frac{5}{5} & = & - k (41.25-24) - - - (2) (∵ R o o m t e m p e r a t u r e i s 24 ° C) \\ s e c o n d p a r t o f q u e s t i o n, g i v e n v a l u e s s u b s t i t u t e i n e q n (1) \\ \frac{80-75}{t_{2}} & = & - k (\frac{80+75}{2} {-θ}_{0}) \\ \frac{5}{t_{2}} & = & - k (38.75-24) - - - (3) \\ e q n (2) \div e q n (3) \\ t_{2} = 5.47 m i n \end{array}$
$t_{2}$ is more than 5 minutes

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