Thermal Diffusivity Definition

Thermal diffusivity is a crucial material property that measures the rate at which heat propagates through a material. It indicates how quickly a material can respond to a change in temperature. In simpler terms, it tells us how fast temperature changes will travel through an object. Materials with high thermal diffusivity can conduct heat quickly, while those with low thermal diffusivity are better at insulating.

Thermal Diffusivity Expression

The expression for thermal diffusivity is derived from the fundamental principles of heat transfer. It relates a material's ability to conduct heat to its ability to store thermal energy. The formula for thermal diffusivity, often denoted by the symbol 'α' (alpha), is given by:

\[ \alpha = \frac{k}{\rho c} \]

Where:

• k is the thermal conductivity of the material. Thermal conductivity measures a material's ability to transfer heat. A higher 'k' means better heat conduction. Its SI unit is Watts per meter Kelvin (\(W/(m \cdot K)\)).
• \(\rho\) (rho) is the density of the material. Density is the mass per unit volume. A higher density means more mass packed into a given volume. Its SI unit is kilograms per cubic meter (\(kg/m^3\)).
• c is the specific heat capacity of the material. Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of the material by one degree Kelvin (or Celsius). A higher 'c' means the material can store more thermal energy. Its SI unit is Joules per kilogram Kelvin (\(J/(kg \cdot K)\)). Note that in some contexts, specific heat capacity might be denoted as \(C_p\) for constant pressure or \(C_v\) for constant volume. In this question, 'c' is used, which commonly refers to specific heat capacity.

Understanding Thermal Diffusivity Components

Let's break down why this expression makes sense:

• The numerator, k (thermal conductivity), represents how well a material conducts heat. A higher 'k' leads to higher thermal diffusivity because heat can spread faster.
• The denominator, \(\rho c\), represents the volumetric heat capacity of the material. This term tells us how much thermal energy a unit volume of the material can store for a given temperature change. It's essentially the material's thermal inertia.
• When 'k' is high and '\(\rho c\)' is low, the material has high thermal diffusivity, meaning heat spreads quickly because it conducts well and doesn't store much heat per unit volume.
• Conversely, when 'k' is low and '\(\rho c\)' is high, the material has low thermal diffusivity, meaning heat spreads slowly because it conducts poorly and stores a lot of heat per unit volume.

Dimensional Analysis for Thermal Diffusivity

To confirm the expression, we can perform a dimensional analysis:

Units of \(k\): \(\frac{W}{m \cdot K} = \frac{J}{s \cdot m \cdot K}\)

Units of \(\rho\): \(\frac{kg}{m^3}\)

Units of \(c\): \(\frac{J}{kg \cdot K}\)

So, the units of \(\frac{k}{\rho c}\) are:

\[ \frac{\frac{J}{s \cdot m \cdot K}}{\frac{kg}{m^3} \cdot \frac{J}{kg \cdot K}} = \frac{\frac{J}{s \cdot m \cdot K}}{\frac{J}{m^3 \cdot K}} = \frac{J}{s \cdot m \cdot K} \cdot \frac{m^3 \cdot K}{J} = \frac{m^2}{s} \]

The unit of thermal diffusivity is \(m^2/s\), which is consistent with the unit of a diffusion coefficient, confirming the correctness of the expression.

Comparing this derived expression with the given options, the correct expression for thermal diffusivity is \(\frac{k}{\rho c}\).