Question

Two identical cylindrical rods except one has a thermal conductivity k1 and the other k2. The rods are placed end to end between the two walls that are 1m apart and maintained at different temperatures, Tw and Tc. T3 the interface temperature, is halfway between the walls. If k2 = 0.5k1 and the heat flows only along the lengths of the rod determine the temperature T3. (Please calculate to final).

Answer 1

given

walls that are 1 m apart

k₂ = k₁ / 2

Q = Heat passes through left to right by a cylinder + Heat flows left to right by other cylinder

then the conductivity equation "H"

H = K x A x ( T₂ - T₁ ) / L

L is same then,

H₁ = k₁ x A x ( T_c - T_w ) / L

H₂ = k₂ x A x ( T_c - T_w ) / L

k₂ = 1/2 k₁

H₂ = 1/2 k₁ x A x ( T_c - T_w ) / L

Q ' = H₁ + H₂

= k₁ x A x ( T_c - T_w ) / L + 1/2 k₁ x A x ( T_c - T_w ) / L

Q ' = 3 x k₁ x A x ( T_c - T_w ) / 2 L

k₁ x A x ( T₃ - T_c ) / L = 1/2 k₁ x A x ( T_w - T₃) / L

( T₃ - T_c ) / L = 1/2 x ( T_w - T₃) / L

2 x ( T₃ - T_c ) / L = ( T_w - T₃) / L  

2 x ( T₃ - T_c ) = ( T_w - T₃)

2 T₃ - 2 T_c = T_w - T

3 T₃ = 2 T_c + T_w

T₃ = ( 2 T_c + T_w ) / 3