Correct Answer - Option 2 : L = 100 cm, r = 2 cm
Concept:
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Thermal Conductivity: When one end of a metal rod is heated, heat flows by conduction from the hot end to the cold end.
- In this process, each cross-section of the rod receives some heat from the adjacent cross-section towards the hot end.
- It is found that the amount of heat Q that flows from hot to cold face during steady-state is
\(Q = \frac{{KA\left( {{T_1} - {T_2}} \right)t}}{L}\)
Where K = Coefficient of thermal conductivity of the material, L is the length.
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Rate of conduction of heat energy is given by:
\(\frac{{dQ}}{t} = \frac{{KA\left( {{T_1} - {T_2}} \right)}}{x} = KA\frac{{{\bf{Δ }}T}}{L}\)
Calculation:
So, the rate of conduction is given as
\(x = KA\frac{{{\bf{Δ }}T}}{L}\)
It is a cylindrical rod, so the cross-sectional area will be given as
\(x = Kπ r^2 \frac{{{\bf{Δ }}T}}{L}\)
Now, for the given temperature change ΔT, K, π are constant. So, the rate depends upon the ratio \(\frac{r^2}{L}\)
If calculate this ratio of all given options,
1: L = 50 cm, r = 1 cm
\(\frac{1^2}{50} = 0.02\)
2: L = 100 cm, r = 2 cm
\(\frac{2^2}{100} = 0.04\)
3: L = 25 cm, r = 0.5 cm
\(\frac{0.5^2}{25} = 0.01\)
4: L = 75 cm, r = 1.5 cm
\(\frac{1.5^2}{75} = 0.03\)
So, the maximum heat transfer will occur with L = 100 cm, r = 2 cm.
