Two Rectangular Blocks, Having Identical Dimensions, Can Be Arrange Either In Configuration I Or In Configuration II. One Of The Blocks Has Thermal Conductivity K And The Other 2k. The Temperature Difference Between The Ends Along The X-axis Is The Same In Both The Configurations. It Takes 9s To Transport A Certain Amount Of Heat From The Hot End To The Cold End In The Configuration I. The Time To Transport The Same Amount Of Heat In The Configuration Ii Is?

Sol:

\( \Omega_{1} = \frac{L}{KA}\\\Omega_{2} = \frac{L}{KA}\\\Omega = \Omega_{1} + \Omega_{2}\\\Rightarrow \frac{1}{KA} + \frac{2}{2KA} = \frac{3}{2}\frac{L}{KA}\\\Rightarrow \Omega = \frac{\Omega_{1}\Omega_{2}}{\Omega_{1} + \Omega_{2}}\\\Rightarrow \frac{\frac{L}{KA} * \frac{L}{2KA}}{\frac{L}{2KA} + \frac{L}{KA}}\\\Rightarrow\frac{\frac{L^{2}}{2K^{2}A^{2}}}{\frac{2L+L}{2KA}}\\\Rightarrow \frac{L^{2}}{2K^{2}A^{2}} * \frac{2KA}{3L} = \frac{L}{3KA}\\\Rightarrow \frac{\Delta Q}{t} = \frac{\Delta T}{\Omega} \\\Rightarrow \frac{t_{i}}{t_{ii}} = \frac{\Omega_{i}}{\Omega_{ii}}\\\Rightarrow \frac{\frac{L}{3KA}}{\frac{3L}{2KA}} * 9 \\\Rightarrow \frac{L}{3KA} * {\frac{2KA}{3L}} * 9 = 2\)

The time to transport the same amount of heat in configuration II is 2 seconds.

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