Understanding the Unit of Thermal Diffusivity

Thermal diffusivity is an important material property that describes how quickly thermal energy diffuses through a material relative to the amount of energy stored within it. It is often denoted by the symbol $\alpha$ (alpha).

The definition of thermal diffusivity is given by the ratio of thermal conductivity to the volumetric heat capacity:

$\alpha = \frac{k}{\rho c_p}$

Where:

• $k$ is the thermal conductivity of the material.
• $\rho$ is the density of the material.
• $c_p$ is the specific heat capacity at constant pressure.

To find the unit of thermal diffusivity, we can analyze the units of the terms in the formula:

• The standard unit for thermal conductivity ($k$) is $\rm \frac{W}{m \cdot K}$, which is equivalent to $\rm \frac{J}{s \cdot m \cdot K}$ (since 1 Watt = 1 Joule/second).
• The standard unit for density ($\rho$) is $\rm \frac{kg}{m^3}$.
• The standard unit for specific heat capacity ($c_p$) is $\rm \frac{J}{kg \cdot K}$.

Now, let's substitute these units into the formula for thermal diffusivity:

$\text{Unit of } \alpha = \frac{\text{Unit of } k}{\text{Unit of } \rho \times \text{Unit of } c_p}$

$\text{Unit of } \alpha = \frac{\frac{J}{s \cdot m \cdot K}}{\frac{kg}{m^3} \times \frac{J}{kg \cdot K}}$

Let's simplify the denominator first:

$\text{Unit of } (\rho c_p) = \frac{kg}{m^3} \times \frac{J}{kg \cdot K} = \frac{J}{m^3 \cdot K}$

Now, substitute this back into the expression for the unit of $\alpha$:

$\text{Unit of } \alpha = \frac{\frac{J}{s \cdot m \cdot K}}{\frac{J}{m^3 \cdot K}}$

$\text{Unit of } \alpha = \frac{J}{s \cdot m \cdot K} \times \frac{m^3 \cdot K}{J}$

Cancel out the units that appear in both the numerator and the denominator (J and K):

$\text{Unit of } \alpha = \frac{1}{s \cdot m} \times m^3 = \frac{m^3}{s \cdot m} = \frac{m^2}{s}$

The standard unit derived is $\rm \frac{m^2}{s}$. However, the options provided use hours (hr) as the time unit instead of seconds (s). Since 1 hour = 3600 seconds, a unit like $\rm \frac{m^2}{hr}$ is also a valid unit for thermal diffusivity, representing the same physical dimension (Area/Time).

Comparing our derived unit $\rm \frac{m^2}{s}$ (which has the same dimensions as $\rm \frac{m^2}{hr}$) with the given options:

• Option 1: $\rm \frac{m^2}{hr}$
• Option 2: $\rm \frac{m^2}{hr.^\circ C}$ (Incorrect, includes temperature)
• Option 3: $\rm \frac{J}{m^2.hr}$ (Incorrect, units of energy flux density)
• Option 4: $\rm \frac{J}{m^2.hr.^\circ C}$ (Incorrect, units related to heat transfer coefficient)

The unit $\rm \frac{m^2}{hr}$ matches the expected dimensions for thermal diffusivity and is one of the common ways to express this unit in practice, especially in engineering contexts where hours are often used for time.

Therefore, the unit of thermal diffusivity is $\rm \frac{m^2}{hr}$.