### Thermal Conductivity – Definition, Derivation, Dimension and Sample Values

##### October 16, 2017

# What is thermal conductivity? – Concepts

**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.

###### Unit

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 = L

^{2})

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}= **(ML ^{2})(T^{-3})**_________(3)

Putting the dimension of Work in equation 2,

**Dimension of Thermal Conductivity (k) = (ML ^{2})(T^{-3}) L^{-1} θ^{-1} = M^{1} L^{1} T ^{-3} θ ^{-1} ** ______ (4)

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

## Some sample values of Thermal Conductivity (k)

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

**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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