Unit of thermal conductivity is:

Question

Prepp Team · Accepted Answer

Understanding the Unit of Thermal Conductivity

Thermal conductivity is a physical property of a material that describes its ability to conduct heat. Materials with high thermal conductivity transfer heat efficiently, while those with low thermal conductivity act as insulators.

To determine the unit of thermal conductivity, we can use Fourier's Law of Heat Conduction. This law describes the rate of heat transfer through a material.

Fourier's Law of Heat Conduction

Fourier's Law states that the rate of heat transfer through a material is proportional to the negative gradient in temperature and the area through which the heat flows. Mathematically, it is often expressed as:

$$ Q = -k A \frac{dT}{dx} $$

Where:

$Q$ is the rate of heat transfer (heat flow per unit time). The unit of $Q$ is Joules per second (J/s), which is also equivalent to Watts (W).
$k$ is the thermal conductivity of the material. This is the property we want to find the unit for.
$A$ is the area through which heat flows. The unit of $A$ is square meters ($m^2$).
$\frac{dT}{dx}$ is the temperature gradient, representing the change in temperature ($dT$) over a change in distance ($dx$). The unit of $dT$ can be Kelvin (K) or degrees Celsius (°C), and the unit of $dx$ is meters (m). So, the unit of $\frac{dT}{dx}$ is K/m or °C/m. Since temperature differences are the same in Kelvin and Celsius scales, K/m is commonly used in SI units.

The negative sign in the formula indicates that heat flows in the direction of decreasing temperature.

Deriving the Unit of Thermal Conductivity

We can rearrange Fourier's Law to solve for $k$:

$$ k = -\frac{Q}{A \frac{dT}{dx}} $$

Now, let's substitute the units of the other quantities into this equation to find the unit of $k$:

$$ \text{Unit of } k = \frac{\text{Unit of } Q}{\text{Unit of } A \times \text{Unit of } \frac{dT}{dx}} $$

$$ \text{Unit of } k = \frac{\text{W}}{\text{m}^2 \times \left(\frac{\text{K}}{\text{m}}\right)} $$

Now, simplify the expression:

$$ \text{Unit of } k = \frac{\text{W}}{\text{m}^2 \times \frac{\text{K}}{\text{m}}} $$

$$ \text{Unit of } k = \frac{\text{W}}{\text{m} \times \text{K}} $$

So, the standard SI unit of thermal conductivity is Watts per meter Kelvin, written as W/(m·K) or W/mK.

Comparing with Options

Let's look at the given options and compare them with our derived unit W/mK:

Option	Unit	Match with W/mK?	Explanation
1	J/m/s	No	J/s is Watt (W). So this is W/m. This unit doesn't include a temperature difference term (K or °C).
2	W/m²K	No	This unit includes area (m²) in the denominator instead of meter (m). It is related to thermal conductance per unit area.
3	W/mK	Yes	This matches our derived unit for thermal conductivity.
4	J/°C	No	J/°C is related to heat capacity, not thermal conductivity. It represents energy required to change temperature by 1 degree Celsius.

Based on the derivation from Fourier's Law, the correct unit for thermal conductivity is W/mK.

Revision Table: Key Heat Transfer Concepts

Concept	Definition	Relevant Formula (Example)	Common Unit
Heat Transfer Rate ($Q$)	Rate at which heat energy is transferred	$Q = \frac{\Delta E}{\Delta t}$	Watt (W) or J/s
Thermal Conductivity ($k$)	Ability of a material to conduct heat	$Q = -k A \frac{dT}{dx}$ (rearranged for Q)	W/mK
Temperature Gradient ($\frac{dT}{dx}$)	Rate of temperature change with distance	N/A	K/m or °C/m
Heat Flux ($q''$)	Rate of heat transfer per unit area	$q'' = \frac{Q}{A} = -k \frac{dT}{dx}$	W/m²
Thermal Resistance ($R_{th}$)	Opposition to heat flow	$Q = \frac{\Delta T}{R_{th}}$	K/W or °C/W

Additional Information on Thermal Conductivity

Thermal conductivity is an important property in many applications, including building insulation, heat sinks in electronics, and design of heat exchangers. Here are some additional points:

Factors Affecting Thermal Conductivity: Thermal conductivity is affected by temperature, pressure, and the phase of the material (solid, liquid, gas). For most materials, thermal conductivity changes with temperature.
Range of Values: Metals typically have high thermal conductivity (e.g., Copper: ~400 W/mK, Aluminum: ~205 W/mK). Non-metals and insulators have much lower values (e.g., Wood: ~0.04 - 0.12 W/mK, Air: ~0.026 W/mK at room temperature).
Relation to Thermal Diffusivity: Thermal conductivity ($k$) is related to thermal diffusivity ($\alpha$) by the formula $\alpha = \frac{k}{\rho c_p}$, where $\rho$ is density and $c_p$ is specific heat capacity at constant pressure. Thermal diffusivity indicates how quickly temperature changes within a material.