A rod of negligible heat capacity has length 20 cm, area of cross section 1.0 cm2 and thermal conductivity 200 W m−1°C−1. The temperature of one end is maintained at 0°C and that of the other end is slowly and linearly varied from 0°C to 60°C in 10 minutes. Assuming no loss of heat through the sides, find the total heat transmitted through the rod in these 10 minutes.
Length of the rod, l = 20 cm = 0.2 m
Area of cross section of the rod, A = 1.0 cm2 = 1.0 × 10 -4m2
Thermal conductivity of the material of the rod, k = 200 W m-1° C-1
The temperature of one end of the rod is increased uniformly by 60° C within 10 minutes.
This mean that the rate of increase of the temperature of one end is 0.1° C per second
`rArr 60/(10xx60)""^circ C//s`
So, total heat flow can be found by adding heat flow every second.
Rate of flow of heat = `(dQ)/dt`
Q"net" = ∑ `(KA)/d ( T_2 - T_1)×Deltat`
For each interval,
`Deltat = 1`
`Q"net" = KA/d ( 0.1 + (KA)/d xx0.2 + (KA)/d xx0.3 + ...................+ KA/dxx 60.0`
`Q"net" = (KA)/d (0.1 + 0.2 +.........+ 60.0)`
sum of n terms of an AP is given by
`s_n = n/2 ( axxa_n)`
`Q_net = (KA)/d xx 600/2 (0.1+60)`
⇒` Q'"net" = (200xx - 10^-4)/(20xx10^-2) xx 600/2xx60.1`
⇒ Qnet = 1800J (approximately)