Question

Thermal diffusivity of a material ($ \alpha $) is defined as:

• A. \( \alpha = \frac{k}{\rho C_p} \)
• B. \( \alpha = \frac{k}{C_p} \)
• C. \( \alpha = \frac{k C_p}{\rho} \)
• D. \( \alpha = \frac{k}{\rho} \)

Hint:

Think of thermal diffusivity as a measure of "temperature agility." A material with high thermal diffusivity will heat up or cool down quickly in response to changes in its surroundings. This happens if it conducts heat well (\( k \) is high) and doesn't require much heat to change its temperature (\( \rho C_p \) is low).

Answer 1

The Correct Option is A

Step 1: Understand the concept of thermal diffusivity. 
Thermal diffusivity (\( \alpha \)) is a measure of how quickly a material can change its temperature in response to a change in the surrounding temperature. It represents the ratio of a material's ability to conduct heat to its ability to store heat. 
Step 2: Recall the definitions of the relevant thermal properties. 
Thermal conductivity (\( k \)): A measure of a material's ability to conduct heat. It quantifies the amount of heat that flows through a unit thickness of the material per unit area per unit temperature difference.
Density (\( \rho \)): Mass per unit volume of the material.
Specific heat capacity at constant pressure (\( C_p \)): The amount of heat required to raise the temperature of a unit mass of the material by one degree Celsius (or Kelvin) at constant pressure.
Step 3: Understand the physical significance of the ratio \( \frac{k}{\rho C_p} \). 
\( k \) in the numerator indicates the rate of heat transfer through the material. A higher \( k \) means heat can propagate quickly.
\( \rho C_p \) in the denominator represents the volumetric heat capacity of the material. It is the amount of heat required to raise the temperature of a unit volume of the material by one degree. A higher \( \rho C_p \) means the material can store more thermal energy for a given temperature change, making its temperature change more sluggish. Therefore, the ratio \( \frac{k}{\rho C_p} \) represents how effectively a material can transfer heat relative to how much heat it can store per unit volume. A high thermal diffusivity means that temperature changes propagate quickly through the material because it conducts heat well and has a relatively low volumetric heat capacity. 
Step 4: Evaluate the given options by comparing them to the definition of thermal diffusivity. 
Option 1: \( \alpha = \frac{k}{\rho C_p} \) - This matches the standard definition of thermal diffusivity.
Option 2: \( \alpha = \frac{k}{C_p} \) - This lacks the density term in the denominator, making the units incorrect for diffusivity (which is \( m^2/s \)).
Option 3: \( \alpha = \frac{k C_p}{\rho} \) - This has specific heat capacity in the numerator, which is contrary to the concept that higher heat capacity leads to lower diffusivity. The units are also incorrect.
Option 4: \( \alpha = \frac{k}{\rho} \) - This lacks the specific heat capacity term, which is essential for relating heat flow to temperature change. The units are also incorrect. 
Step 5: Select the correct answer. 
The thermal diffusivity (\( \alpha \)) of a material is correctly defined as \( \alpha = \frac{k}{\rho C_p} \).