SATHEE: Heat and Thermodynamics 6 Question 26

Heat and Thermodynamics 6 Question 26

31. One end of a rod of length $L$ and cross-sectional area $A$ is kept in a furnace of temperature $T_{1}$ . The other end of the rod is kept at a temperature $T_{2}$ . The thermal conductivity of the material of the $rod$ is $K$ and emissivity of the $rod$ is $e$ .

It is given that $T_{2} = T_{s} + Δ T$ , where $Δ T « T_{s}, T_{s}$ being the temperature of the surroundings. If $Δ T \propto (T_{1} - T_{s})$ , find the proportionality constant. Consider that heat is lost only by radiation at the end where the temperature of the $rod$ is $T_{2}$ .

(2004, 4M)

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Answer:

Correct Answer: 31. Proportionality constant $= \frac{K}{4 e σ L T_{s}^{3} + K}$

Solution:

Rate of heat conduction through rod $=$ rate of the heat lost from right end of the rod.

$∴ \frac{K A (T_{1} - T_{2})}{L} = e A σ (T_{2}^{4} - T_{s}^{4}) \dots (i)$

Given that

$\begin{aligned} T_{2} & = T_{s} + Δ T \\ T_{2}^{4} & = {(T_{s} + Δ T)}^{4} \\ = T_{s}^{4} {(1 + \frac{Δ T}{T_{s}})}^{4} \end{aligned}$

$∴ T_{2}^{4} = {(T_{s} + Δ T)}^{4}$

Using binomial expansion, we have

$\begin{aligned} T_{2}^{4} & = T_{s}^{4} (1 + 4 \frac{Δ T}{T_{s}}) (as Δ T < T_{s}) \\ ∴ T_{2}^{4} - T_{s}^{4} & = 4 (Δ T) (T_{s}^{3}) \end{aligned}$

Substituting in Eq. (i), we have

$\begin{aligned} \frac{K (T_{1} - T_{s} - Δ T)}{L} & = 4 e σ T_{s}^{3} \cdot Δ T \\ or & \frac{K (T_{1} - T_{s})}{L} & = (4 e σ T_{s}^{3} + \frac{K}{L}) Δ T \\ ∴ & Δ T & = \frac{K (T_{1} - T_{s})}{(4 e σ L T_{s}^{3} + K)} \end{aligned}$

Comparing with the given relation, proportionality constant $= \frac{K}{4 e σ L T_{s}^{3} + K}$

Heat and Thermodynamics 6 Question 26

31. One end of a rod of length $L$ and cross-sectional area $A$ is kept in a furnace of temperature $T_{1}$ . The other end of the rod is kept at a temperature $T_{2}$ . The thermal conductivity of the material of the $rod$ is $K$ and emissivity of the $rod$ is $e$ .

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Heat and Thermodynamics 6 Question 26

31. One end of a rod of length L and cross-sectional area A is kept in a furnace of temperature T1. The other end of the rod is kept at a temperature T2. The thermal conductivity of the material of the rod is K and emissivity of the rod is e.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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31. One end of a rod of length $L$ and cross-sectional area $A$ is kept in a furnace of temperature $T_{1}$ . The other end of the rod is kept at a temperature $T_{2}$ . The thermal conductivity of the material of the $rod$ is $K$ and emissivity of the $rod$ is $e$ .