Question

Prandtl number is the ratio of ______

• A. thermal diffusivity to mass diffusivity
• B. momentum diffusivity to thermal diffusivity
• C. mass diffusivity to thermal diffusivity
• D. thermal diffusivity to momentum diffusivity

Hint:

Think of dimensionless numbers as transport ratios:
Prandtl Number (\( Pr \)) = Momentum Diffusivity / Thermal Diffusivity
Schmidt Number (\( Sc \)) = Momentum Diffusivity / Mass Diffusivity
Lewis Number (\( Le \)) = Thermal Diffusivity / Mass Diffusivity
These three numbers form a family of transport property ratios.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the physical definition of the dimensionless Prandtl number (\( Pr \)).
The Prandtl number is a fundamental parameter in transport phenomena that links fluid dynamics and heat transfer.

Step 2: Key Formula or Approach:
The Prandtl number is mathematically defined as:
\[ Pr = \frac{\nu}{\alpha} \]
where \( \nu \) represents the kinematic viscosity (or momentum diffusivity) and \( \alpha \) represents the thermal diffusivity.

Step 3: Detailed Explanation:

• Momentum Diffusivity (\( \nu \)): Kinematic viscosity represents the rate of momentum transport through molecular friction in fluid flow:
\[ \nu = \frac{\mu}{\rho} \]
where \( \mu \) is dynamic viscosity and \( \rho \) is fluid density.

• Thermal Diffusivity (\( \alpha \)): This measures the rate of heat transport through conduction in the material:
\[ \alpha = \frac{k}{\rho \cdot C_p} \]
where \( k \) is thermal conductivity and \( C_p \) is specific heat capacity.

• Combining the ratios:
\[ Pr = \frac{\nu}{\alpha} = \frac{\frac{\mu}{\rho}}{\frac{k}{\rho \cdot C_p}} = \frac{\mu \cdot C_p}{k} \]
This ratio represents the relative thickness of the momentum boundary layer and the thermal boundary layer.

• Physical significance:
If \( Pr \approx 1 \) (e.g., gases), the momentum and thermal boundary layers grow at the same rate.
If \( Pr \gg 1 \) (e.g., oils), momentum diffuses much faster than heat, meaning the velocity boundary layer is much thicker than the thermal boundary layer.


Step 4: Final Answer:
The Prandtl number is defined as the ratio of momentum diffusivity to thermal diffusivity.