Thermal Conductivity Derivation| Dimension of thermal conductivity

The thermal conductivity of a material (designated as k) is the measure of how quickly heat energy is transferred or conducted from one end of the material to the other end. In this post we will work on the derivation of the thermal conductivity formula first,  then we will find the dimension of thermal conductivity as well.

Derivation of Thermal Conductivity Expression

Now we will derive the Thermal Conductivity expression.

Suppose heat energy Q is flowing through a rod of length L in time t.

The temperature values of the two ends of the rod are T1 and T2. (Say, T1>T2)

Then the Rate of Flow of Heat i.e. Q/t, through the rod in the steady-state is:

-Proportional to the cross-sectional area A of the rod and
-Proportional to the temperature difference (T1-T2) between the two ends of the rod
-and is Inversely proportional to the length or thickness (L) of the rod.

Q/t ∝  [A (T1-T2)]/L

=>     Q/t = [k A (T1-T2)] / L,

where k is a constant called the thermal conductivity of the material of the rod.

=> k = [Q L] / [A (T1-T2) t ]    …………………… (1)

Unit

SI unit of k is   Watt meter-1 Kelvin-1 

Dimension of Thermal Conductivity

Here we will find out the Dimension of k.

From equation 1, we can clearly see that

k = (Q/t).L.A-1(T1-T2)-1

From this equation, we will gradually derive the dimension. Let’s read on to get it.
And we know, Dimension of Q/t is equal to the dimension of Work/time or i.e. Power.

The dimension of L.A-1 is equal to that of L-1 actually. (as A = L2)

Temperature difference (T1-T2) can be designated with Theta (θ)

So we can write the expression in this way,
k = (Unit of Power) (unit of length)-1(unit of temperature)-1 _____________ (2)

Breaking these down for simplicity,

Power = Work/time = (force X displacement) / time

= (mass X acceleration X Displacement)time-1

So, Power = M (LT-2) L T-1= (ML2)(T-3)_________(3)

Putting the dimension of Work in equation 2,

Dimension of Thermal Conductivity (k)
= (ML2)(T-3) L-1 θ-1 =  M1 L1  T -3  θ -1   ______ (4)

In the next part of this tutorial, let’s find out the values of k for a few selected materials. Link here: Thermal conductivity sample values of selected materials

Suggested READING:

Thermal conductivity significance & sample values of selected materials

thermal conductivity
thermal conductivity
Thermal Conductivity Derivation| Dimension of thermal conductivity
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