Topic: Heat Transfer
The heat flow rate through parallel walls of thickness $L_{1},$ $L_{2}$ and $L_{3},$ having surface areas $A_{1},$ $A_{2}$ and $A_{3}$ thermal conductivities $k_{1},$ $k_{2}$ and $k_{3}$ respectively, with the first and last walls maintained at temperatures $t_{1}$ and $t_{2}$ will be
- A. $\frac{t_{1}-t_{2}}{\frac{L_{1}}{A_{1}k_{1}}+\frac{L_{2}}{A_{2}k_{2}}+\frac{L_{3}}{A_{3}k_{3}}}$
- B. $\frac{t_{1}-t_{2}}{\frac{k_{1}}{A_{1}L_{1}}+\frac{k_{2}}{A_{2}L_{2}}+\frac{k_{3}}{A_{3}L_{3}}}$
- C. $\frac{t_{1}-t_{2}}{\frac{k_{1}A_{1}}{L_{1}}+\frac{k_{2}A_{2}}{L_{2}}+\frac{k_{3}A_{3}}{L_{3}}}$
- D. $\frac{t_{1}-t_{2}}{\frac{L_{1}A_{1}}{k_{1}}+\frac{L_{2}}{k_{2}}+\frac{L_{3}}{k_{3}}}$
Answer: A. $\frac{t_{1}-t_{2}}{\frac{L_{1}}{A_{1}k_{1}}+\frac{L_{2}}{A_{2}k_{2}}+\frac{L_{3}}{A_{3}k_{3}}}$
Explanation: For a composite wall made of three parallel layers, the total thermal resistance is the sum of the individual resistances: $R_{th} = \frac{L_{1}}{k_{1}A_{1}} + \frac{L_{2}}{k_{2}A_{2}} + \frac{L_{3}}{k_{3}A_{3}}$. The heat transfer rate $Q = \frac{dT}{R_{th}} = \frac{t_{1}-t_{2}}{\frac{L_{1}}{k_{1}A_{1}}+\frac{L_{2}}{k_{2}A_{2}}+\frac{L_{3}}{k_{3}A_{3}}}$.