Question

Which of the following dimensionless number is associated with free convection?

• A. Prandtl number
• B. Grashof number
• C. Peclet number
• D. Reynolds number

Hint:

For natural convection problems, the transition from laminar to turbulent flow is determined by the Rayleigh number (\( Ra \)), which is the product of the Grashof and Prandtl numbers:
\[ Ra = Gr \cdot Pr \]
Typically, a transition occurs around \( Ra \approx 10^9 \) for a vertical plate.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks which dimensionless number is used to characterize natural (free) convection heat transfer.
Free convection occurs when fluid motion is driven entirely by buoyancy forces resulting from density differences caused by temperature gradients.

Step 2: Key Formula or Approach:
The Grashof number (\( Gr \)) acts as the equivalent of the Reynolds number for free convection. It is defined as:
\[ Gr = \frac{g \cdot \beta \cdot (T_s - T_{\infty}) \cdot L^3}{\nu^2} \]
where:
\( g \) is acceleration due to gravity,
\( \beta \) is the volumetric thermal expansion coefficient,
\( T_s \) is the surface temperature,
\( T_{\infty} \) is the ambient fluid temperature,
\( L \) is the characteristic length,
\( \nu \) is the kinematic viscosity.

Step 3: Detailed Explanation:

• Grashof Number: The physical interpretation of the Grashof number is the ratio of buoyancy forces to viscous forces acting on the fluid:
\[ Gr = \frac{\text{Buoyant Force}}{\text{Viscous Force}} \]
Since buoyancy drives the fluid flow in free convection, \( Gr \) is the primary dimensionless number used to determine if the boundary layer flow is laminar or turbulent.

• Other Options:
-

Reynolds number (\( Re \)): Represents the ratio of inertial forces to viscous forces. It is the primary dimensionless number used to characterize forced convection.
-

Prandtl number (\( Pr \)): Represents the ratio of momentum diffusivity to thermal diffusivity. It is a fluid property parameter used in both free and forced convection.
-

Peclet number (\( Pe \)): Represents the ratio of advective transport to diffusive transport (\( Pe = Re \cdot Pr \)).


Step 4: Final Answer:
The Grashof number is the dimensionless parameter associated with free convection.