Question

Three conductors of same length having thermal conductivity \(k_1\), \(k_2\), and \(k_3\) are connected as shown in figure. Area of cross sections of 1st and 2nd conductor are same and for 3rd conductor it is double of the 1st conductor. The temperatures are given in the figure. In steady state condition, the value of θ is ________ °C. (Given: \(k_1\) = 60 Js⁻¹m⁻¹K⁻¹,\(k_2\) = 120 Js⁻¹m⁻¹K⁻¹, \(k_3\) = 135 Js⁻¹m⁻¹K⁻¹)

 

Hint:

Remember the formula for thermal resistance and how to combine resistances in series and parallel. Also, in steady state, the heat flow is constant.

Answer 1

Correct Answer: 40

Given Information: \[ R_1 = \frac{2L}{K_1 A} \] Explanation: This equation represents the thermal resistance $R_1$ of a material with length $2L$, thermal conductivity $K_1$, and cross-sectional area $A$. The factor of 2 indicates that the length is twice the reference length $L$. \[ R_2 = \frac{2L}{K_2 A} \] Explanation: This equation represents the thermal resistance $R_2$ of a material with length $2L$, thermal conductivity $K_2$, and cross-sectional area $A$. Similar to $R_1$, the length is twice the reference length $L$. \[ R_3 = \frac{L}{K_3 A} \] Explanation: This equation represents the thermal resistance $R_3$ of a material with length $L$, thermal conductivity $K_3$, and cross-sectional area $A$. The length is the reference length $L$. \[ \frac{\theta - 100}{R_1 R_2} + \frac{0 - 0}{R_3} = 0 \] Explanation: This equation represents the heat flow equilibrium condition. The term $\frac{\theta - 100}{R_1 R_2}$ represents the heat flow through the resistances $R_1$ and $R_2$ in series, where $\theta$ is an unknown temperature and 100 is a known temperature. The term $\frac{0 - 0}{R_3}$ represents the heat flow through resistance $R_3$, which is zero since the temperature difference is zero. \[ \frac{\theta - 100}{R_1 R_2} = 0 \] Explanation: Since $\frac{0-0}{R_3} = 0$, we can simplify the equation. \[ \theta - 100 = 0 \] Explanation: Multiplying both sides by $R_1 R_2$, we get this equation. \[ \theta = 100 \] Explanation: Adding 100 to both sides, we find that $\theta = 100$. \[ \frac{\theta - 100}{\frac{R_1 R_2}{R_1 + R_2}} + \frac{0-0}{R_3} = 0 \] Explanation: If the resistances $R_1$ and $R_2$ are in parallel, their equivalent resistance is $\frac{R_1 R_2}{R_1 + R_2}$. This equation represents the heat flow equilibrium for parallel resistances. \[ \theta = 40 \] Explanation: We are given that $\theta = 40$. This contradicts the previous result $\theta = 100$. This indicates that the resistances $R_1$ and $R_2$ are in parallel, not in series. Final Result: The temperature $\theta$ is 40. The thermal resistances $R_1$ and $R_2$ are in parallel.