In the study of heat transfer, natural convection is a mode of heat transfer that occurs due to density differences in a fluid (liquid or gas) caused by temperature gradients. When a fluid is heated, it becomes less dense and rises, while cooler, denser fluid sinks. This movement of fluid driven by buoyancy forces is what facilitates heat transfer in natural convection.

Understanding Dimensionless Numbers in Natural Convection

To analyze natural convection heat transfer, we use several dimensionless numbers. These numbers help us characterize the fluid flow and heat transfer regime without depending on the specific size or properties of the system. The Nusselt number is a key parameter used to quantify the convective heat transfer.

• Nusselt number (Nu): This number represents the ratio of convective heat transfer to conductive heat transfer across a boundary. A higher Nusselt number indicates more effective convection compared to conduction. Mathematically, $\text{Nu} = \frac{h L}{k}$, where $h$ is the convective heat transfer coefficient, $L$ is the characteristic length, and $k$ is the thermal conductivity of the fluid.
• Reynolds number (Re): This number is the ratio of inertial forces to viscous forces in a fluid. It is primarily used to predict flow patterns in different fluid flow situations, particularly in forced convection. $\text{Re} = \frac{\rho v L}{\mu}$.
• Prandtl number (Pr): This number relates the momentum diffusivity (kinematic viscosity) to the thermal diffusivity of a fluid. It provides insight into the relative thickness of the momentum boundary layer and the thermal boundary layer. $\text{Pr} = \frac{\nu}{\alpha} = \frac{\mu c_p}{k}$. This is a fluid property.
• Grashof number (Gr): This number is the ratio of buoyancy forces to viscous forces in a fluid. It is the primary parameter that drives natural convection flow. It is analogous to the Reynolds number for forced convection but is based on buoyancy instead of external velocity. $\text{Gr}_L = \frac{g \beta \Delta T L^3}{\nu^2}$.
• Rayleigh number (Ra): This number is the product of the Grashof number and the Prandtl number, i.e., $\text{Ra} = \text{Gr} \cdot \text{Pr}$. It is also a significant dimensionless number in natural convection and is often used directly in correlations.

Nusselt Number Dependence in Natural Convection Heat Transfer

In natural convection, the fluid motion is driven by buoyancy forces, which are characterized by the Grashof number (Gr). The rate of heat transfer is also influenced by the fluid's properties related to momentum and thermal diffusion, which are captured by the Prandtl number (Pr).

Therefore, the Nusselt number in natural convection is fundamentally a function of the Grashof number and the Prandtl number.

The functional relationship is generally expressed as:

$\text{Nu} = f(\text{Gr}, \text{Pr})$

Sometimes, this relationship is expressed in terms of the Rayleigh number, which is the product of Grashof and Prandtl numbers:

$\text{Nu} = f(\text{Ra})$

While the Reynolds number is crucial for analyzing forced convection (where an external force like a pump or fan causes fluid motion), it is not the primary parameter governing the fluid flow in natural convection. The flow in natural convection arises internally from density variations, characterized by the Grashof number.

Conclusion

Based on the principles of natural convection and the role of dimensionless numbers, the Nusselt number, which quantifies convective heat transfer, is determined by the interplay of buoyancy forces (represented by the Grashof number) and the relative rates of momentum and thermal diffusion in the fluid (represented by the Prandtl number). Thus, the Nusselt number in natural convection is a function of the Prandtl number and Grashof number.