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Seven identical rods of material of thermal conductivityK are connected as shown in figure. All the rods are of identical length L and cross sectional area A_1. If one end A is kept at 100^@C and the other end B is kept at 0^@C, what would be the temperature of the junctions C, D and E (T_C,T_D and T_E) in the steady state? |
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Answer» `T_C gt T_E gt T_D`  Let `H_1,H_2,H_3,H_4,H_5,H_6` and `H_7` be the rate of heat flow through AE,AC,CD,CE,EB,ED and DB respectively. Then `H_1=(KA_1(100-T_E))/L ,H_2 = (KA_1(100-T_C))/L` `H_3=(KA_1(T_C-T_D))/L , H_4=(KA_1 (T_C-T_E))/L` `H_5=(KA_1(T_E-0))/L , H_6=(KA_1(T_E-T_D))/L` `H_7=(KA_1(T_D-0))/L because H_1=H_5` `therefore (KA_1(100-T_E))/L =(KA_1(T_E-0))/L` `T_E=50^@ C`...(i) `becauseH_4=H_6` `therefore (KA_1(T_C-50))/L =(KA_1(50-T_D))/L` (using (i)) `T_C+T_D=100` ...(ii)`because H_2=H_3+H_4=H_7` `therefore (KA_1(100-T_C))/L=(KA_1(T_C-T_D))/L + (KA_1(T_C-50))/L=(KA_1(T_D))/L` (Using (i)) `2T_C-2T_D=50` ...(iii) Solving (ii) and (iii) ,we get `T_C=62.5^@C, T_D=37.5^@C rArr T_C gt T_E gt T_D` |
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