Question

Two conductors having the same width and length, thickness ${d_1}$ and ${d_2}$ and thermal conductivity ${k_1}$ and ${k_2}$ are placed one above the other. Find the equivalent thermal conductivity?
$
  {\text{A}}{\text{. }}\dfrac{{\left( {{d_1} + {d_2}} \right)\left( {{k_1}{d_2} + {k_2}{d_1}} \right)}}{{2\left( {{k_1} + {k_2}} \right)}} \\
  {\text{B}}{\text{. }}\dfrac{{\left( {{d_1} - {d_2}} \right)\left( {{k_1}{d_2} + {k_2}{d_1}} \right)}}{{2\left( {{k_1} + {k_2}} \right)}} \\
  {\text{C}}{\text{. }}\dfrac{{{k_1}{d_1} + {k_2}{d_2}}}{{{d_1} + {d_2}}} \\
  {\text{D}}{\text{. }}\dfrac{{{k_1} + {k_2}}}{{{d_1} + {d_2}}} \\
 $

Answer 1

Hint: When two conductors are connected in parallel to each other than the total heat energy stored is equal to the sum of heat energy of each conductor. Then we can use the expression for heat energy in terms of temperature difference and the thermal resistance. Solving this expression for equivalent thermal conductivity, we can get the required answer.
Formula used:
The thermal resistance of a conductor is given as
$R = \dfrac{L}{{kdw}}$
The amount of heat through a conductor is given as a ratio of temperature difference and thermal resistance.
$Q = \dfrac{{\Delta T}}{R}$

Complete answer:
We are given two conductors which are joined parallel to each other as shown in the diagram. Let w be the width of each conductor and L be the length of each conductor.

Let ${Q_1}$ be the amount of heat energy flowing through the conductor 1 and ${Q_2}$ be the amount of heat energy flowing through the conductor 2 then since the conductors are joined in parallel then the total amount of heat is given as
$Q = {Q_1} + {Q_2}$
Let the one end of the combination be at temperature ${T_1}$ and the other at temperature ${T_2}$. Inserting the value of heat energy in terms of the temperature difference and thermal resistance in the above expression we get
$\dfrac{{{T_1} - {T_2}}}{{\dfrac{L}{{{k_{eq}}\left( {{d_1} + {d_2}} \right)w}}}} = \dfrac{{{T_1} - {T_2}}}{{\dfrac{L}{{{k_1}{d_1}w}}}} + \dfrac{{{T_1} - {T_2}}}{{\dfrac{L}{{{k_2}{d_2}w}}}}$
Here ${k_{eq}}$ is the equivalent thermal conductivity of the two given conductors which we need to find out. Solving the above expression, we get
$
  {k_{eq}}\left( {{d_1} + {d_2}} \right) = {k_1}{d_1} + {k_2}{d_2} \\
  {k_{eq}} = \dfrac{{{k_1}{d_1} + {k_2}{d_2}}}{{{d_1} + {d_2}}} \\
 $

So, the correct answer is “Option C”.

Note:
It should be noted from the expression of equivalent thermal conductivity that the equivalent thermal conductivity does depend only on the thermal conductivity of conductors and their thickness. It is independent of the length and the width of the conductors.