Question 1: Thermodynamics & KTG

Question

Two rectangular blocks, having identical dimensions, can be arranged in either configuration-I or configuration-II as shown in the figure. One of the blocks has thermal conductivity $\kappa $ and the other 2$\kappa $. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes 9 s to transport a certain amount of heat from the hot end to the cold end in configuration-I. The time to transport the same amount of heat in configuration-II is
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l1rn03t1/dbcebfcb-4750-4547-831e-4f9923909d41/f3c82750-b7e4-11ec-aca9-51ae2c04643b/file-1l1rn03t2.png?format=png)

Accepted Answer

Let L and A be length and area of cross-section of each block.
$	herefore$ Thermal resistance of block 1 is
$	herefore$ ${R_1} = {L \over {\kappa A}}$
Thermal resistance of block 2 is
${R_2} = {L \over {2\kappa A}}$
In configuration I, two blocks are connected in series. So, their equivalent thermal resistance is
${R_S} = {R_1} + {R_2} = {L \over {\kappa A}} + {L \over {2\kappa A}} = {3 \over 2}{L \over {\kappa A}}$ ...... (i)
$	herefore$ Rate of heat flow in configuration I is
$	herefore$ ${Q \over t} = {{{T_1} - {T_2}} \over {{R_s}}}$ ..... (ii)
In configuration II, two blocks are connected in parallel. So, their equivalent thermal resistance is
${1 \over {{R_P}}} = {1 \over {{R_1}}} + {1 \over {{R_2}}} = {1 \over {{L \over {\kappa A}}}} + {1 \over {{L \over {2\kappa A}}}} = {{3\kappa A} \over L}$ ...... (iii)
${R_P} = {1 \over 3}{L \over {\kappa A}}$
$	herefore$ Rate of same amount of heat flow in configuration II is
$	herefore$ ${Q \over {t'}} = {{{T_1} - {T_2}} \over {{R_P}}}$ ...... (iv)
Divide (ii) by (iv), we get
${{t'} \over t} = {{{R_P}} \over {{R_S}}} = {1 \over 3}{L \over {\kappa A}} 	imes {{2\kappa A} \over {3L}} = {2 \over 9}$ (Using (i) and (iii))
$t' = {2 \over 9}t = {2 \over 9} 	imes 9s = 2s$
![](['https://app-content.cdn.examgoal.net/fly/@width/image/1l1rn2vbd/c8164ae7-acbd-4523-8ea5-928c29674d6c/40a6a6a0-b7e5-11ec-aca9-51ae2c04643b/file-1l1rn2vbe.png?format=png', 'https://app-content.cdn.examgoal.net/fly/@width/image/1l1rn4y5p/bc41837e-1ac6-4550-b5e1-0da8f3dcd96e/7a76e9d0-b7e5-11ec-aca9-51ae2c04643b/file-1l1rn4y5q.png?format=png'])

Answer

2.0 s

Answer

3.0 s

Answer

4.5 s

Answer

6.0 s

Thermodynamics & KTG

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