Thermal Conductivity – Definition, Derivation, Dimension and Sample Values | What is Thermal Conductivity?

What is thermal conductivity? 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 thermal conductivity  expression first,  then we will find its dimension as well.

A material with high k value will conduct heat faster than a material with lower value of k.

If a steady temperature difference is to be maintained between the ends of say, a rod, then heat energy must be supplied at one end of the rod and the same heat must be taken out at the other end of the rod.

Here comes the importance of the ‘capability of heat transfer’ of the material. Materials of high k are used as heat sinks and materials of low k are used as thermal insulation.

thermal conductivity
thermal conductivity

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 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)

Definition of Thermal conductivity from its expression

From its expression in the previous paragraph we can define Thermal conductivity.

When A = 1, t = 1, Temperature difference (T1-T2) =1 and L=1, then k = Q

Thermal Conductivity (k) is the quantity of heat transmitted due to a unit temperature difference between 2 ends of a conductor of unit length(or unit thickness), in unit time under steady conditions in a direction normal to a surface of the unit area.


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.

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 section, let’s find out values of k for a few selected materials.

All values in the list below are at 20 degree Celsius temperature, and unit is Watt per meter per Kelvin

Some sample values of k

Here is a list of Thermal Conductivity values

Pure Silver 407

pure Copper 386

Gold 315

Pure Iron 73

Pure Aluminum 204

Mercury 8.4
Ref: engineeringtoolbox

Suggested READING: Thermodynamics

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Thermal Conductivity – Definition, Derivation, Dimension and Sample Values | What is Thermal Conductivity?
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