Nusselt number and Reynolds number in laminar and turbulent flows

The Nusselt number (Nu) and Reynolds number (Re) are two important dimensionless parameters used in fluid mechanics and heat transfer analysis. The Nusselt number represents the convective heat transfer, while the Reynolds number characterizes the flow regime.

Definition of Nusselt number:
The Nusselt number is a dimensionless parameter used to quantify the convective heat transfer between a fluid and a solid surface. It is defined as the ratio of convective heat transfer to conductive heat transfer:

Nu = hL / k

where:
- Nu is the Nusselt number,
- h is the convective heat transfer coefficient,
- L is a characteristic length scale (such as the diameter of a pipe),
- k is the thermal conductivity of the fluid.

Definition of Reynolds number:
The Reynolds number is a dimensionless parameter used to determine the flow regime (laminar or turbulent) of a fluid. It is defined as the ratio of inertial forces to viscous forces:

Re = ρVL / μ

where:
- Re is the Reynolds number,
- ρ is the density of the fluid,
- V is the velocity of the fluid,
- L is a characteristic length scale (such as the diameter of a pipe),
- μ is the dynamic viscosity of the fluid.

Relationship between Nusselt number and Reynolds number:
The relationship between the Nusselt number and Reynolds number depends on the flow regime (laminar or turbulent).

Laminar flow:
In laminar flow, the fluid moves in smooth, parallel layers with low velocities. For laminar flow, the relationship between the Nusselt number and Reynolds number is given by:

Nu = 0.332Re^0.5Pr^1/3

where Pr is the Prandtl number (a dimensionless parameter representing the ratio of momentum diffusivity to thermal diffusivity).

In this case, the Nusselt number is proportional to the square root of the Reynolds number raised to the power of 0.5. Therefore, option 'B' (Re^0.5) is the correct answer for the relationship between the Nusselt number and Reynolds number in laminar flow.

Turbulent flow:
In turbulent flow, the fluid moves irregularly with high velocities and turbulent eddies. For turbulent flow, the relationship between the Nusselt number and Reynolds number is given by:

Nu = 0.037Re^0.8Pr^1/3

In this case, the Nusselt number is proportional to the Reynolds number raised to the power of 0.8. Therefore, option 'B' (Re^0.8) is also the correct answer for the relationship between the Nusselt number and Reynolds number in turbulent flow.

Conclusion:
In summary, the Nusselt number is related to the Reynolds number in laminar and turbulent flows as Re^0.5 and Re^0.8, respectively. These relationships are derived from empirical correlations and are widely used in heat transfer calculations and engineering design.