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.
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.
In this post we will work on the derivation of thermal conductivity expression first, then we will find its dimension 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 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 Thermal Conductivity.
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 Thermal Conductivity= (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
Pure Silver 407
pure Copper 386
Pure Iron 73
Pure Aluminum 204
Suggested READING: Thermodynamics
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