In finding Reynolds number, the characteristic length of a circular pipe of diameter d is taken as 2d. This is because the Reynolds number is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in a flowing fluid.

Understanding Reynolds Number:
Reynolds number (Re) is used to determine the flow regime of a fluid, whether it is laminar or turbulent. It is calculated using the formula:

Re = (ρ * v * L) / μ

where:
- ρ is the density of the fluid
- v is the velocity of the fluid
- L is the characteristic length of the flow
- μ is the dynamic viscosity of the fluid

Explanation of the Correct Answer:
The characteristic length (L) in the Reynolds number calculation is a measure of the size or scale of the flow geometry. For a circular pipe, the characteristic length can be defined in different ways, such as the pipe diameter (d), the hydraulic diameter (4 times the cross-sectional area divided by the wetted perimeter), or the pipe length.

In this case, the correct answer is option '2d'. This means that the characteristic length is taken as twice the diameter of the circular pipe. This choice is based on the fact that the diameter is a representative length scale for a pipe.

Reasoning behind Option '2d':
The reason for choosing the characteristic length as 2d can be understood by considering the flow behavior in a circular pipe. In laminar flow, the fluid flows in smooth, parallel layers with little mixing between them. In this case, the flow behavior is primarily determined by the pipe diameter itself. Hence, it makes sense to take the characteristic length as 2d, as it represents the size of the pipe.

On the other hand, in turbulent flow, the fluid undergoes chaotic motion with significant mixing and eddies. The flow behavior is influenced by factors such as the pipe roughness and length. However, even in turbulent flow, the pipe diameter still plays a significant role in determining the flow characteristics. Therefore, taking the characteristic length as 2d is a reasonable choice.

In conclusion, the characteristic length of a circular pipe of diameter d is taken as 2d when calculating the Reynolds number because the diameter is a representative length scale for the pipe, and it influences the flow behavior in both laminar and turbulent flow regimes.