The momentum correction factor and the energy correction factor are used to account for the frictional losses in a laminar flow in a pipe. The ratio of these two factors can be determined by considering the equations used to calculate them.

Momentum Correction Factor:
The momentum correction factor, also known as the Darcy-Weisbach friction factor, is used to calculate the head loss due to friction in a pipe. It is given by the equation:

f = 64/Re

where f is the friction factor and Re is the Reynolds number. The Reynolds number is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in a fluid flow. It is given by the equation:

Re = ρVD/μ

where ρ is the density of the fluid, V is the velocity of the fluid, D is the diameter of the pipe, and μ is the dynamic viscosity of the fluid.

Energy Correction Factor:
The energy correction factor, also known as the Fanning friction factor, is used to calculate the pressure drop due to friction in a pipe. It is given by the equation:

f = 4f'

where f' is the Darcy-Weisbach friction factor.

Ratio of Momentum Correction Factor to Energy Correction Factor:
To determine the ratio of the momentum correction factor to the energy correction factor, we can substitute the equation for the momentum correction factor into the equation for the energy correction factor:

f = 4(64/Re)

Simplifying this equation, we find:

f = 256/Re

This shows that the energy correction factor is four times the momentum correction factor. Therefore, the ratio of the momentum correction factor to the energy correction factor is 1/4.

However, it is important to note that in the laminar flow regime, the friction factor is not constant and varies with the Reynolds number. The equations provided above are valid for laminar flows with Reynolds numbers less than 2000. In turbulent flows, different equations are used to calculate the friction factor and the ratio of the momentum correction factor to the energy correction factor may be different.