Question

In free convection, which dimensionless number represents the ratio of buoyancy forces to viscous forces?

• A. Reynolds Number
• B. Grashof Number
• C. Prandtl Number
• D. Nusselt Number

Hint:

Always remember:
- Reynolds ($Re$) $\to$ Forced Convection (Inertia Viscous)
- Grashof ($Gr$) $\to$ Free Convection (Buoyancy Viscous)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks for the dimensionless number that governs free (natural) convection and represents the ratio of buoyant forces to viscous forces.

Step 2: Key Formula or Approach:
The dimensionless number that plays the same role in free convection as the Reynolds number does in forced convection is the Grashof number ($Gr$):
\[ Gr = \frac{g \beta (T_s - T_{\infty}) L^3}{\nu^2} \]

Step 3: Detailed Explanation:

• In free convection, fluid motion is induced by buoyancy forces arising from density differences caused by temperature gradients.

• The Grashof number ($Gr$) is defined physically as:
\[ Gr = \frac{\text{Buoyancy Forces}}{\text{Viscous Forces}} \]

• In this formula:
$g$ is the acceleration due to gravity,
$\beta$ is the volumetric thermal expansion coefficient,
$T_s$ and $T_{\infty}$ are the surface and ambient temperatures,
$L$ is the characteristic length, and
$\nu$ is the kinematic viscosity.

• The Reynolds number represents the ratio of inertia forces to viscous forces.

• The Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity.

• The Nusselt number represents the ratio of convective to conductive heat transfer.

Step 4: Final Answer:
The Grashof number represents the ratio of buoyancy forces to viscous forces in free convection.