Momentum Correction Factor for Flow in Pipes with Variable Velocity Profile

Introduction:
Momentum correction factor is defined as the ratio of the actual force exerted on a fluid by the pipe wall to the force that would be exerted on the fluid if the velocity profile were uniform across the pipe cross-section.

Given Information:
In this question, the velocity is zero over half of the cross-sectional area and is uniform over the remaining half.

Calculation:
Momentum correction factor is given by the formula:

ΔP_actual = fρLV^2/2D

ΔP_ideal = ρV^2/2

f = (ΔP_actual/ΔP_ideal) * (D/L)

where
ΔP_actual is the actual pressure drop in the pipe
ΔP_ideal is the pressure drop for a fluid with uniform velocity profile
f is the friction factor
ρ is the density of the fluid
L is the length of the pipe
D is the diameter of the pipe
V is the average velocity of the fluid

For a pipe with variable velocity profile, the friction factor is given by the equation:

f = (1/2) * [ln⁡(1 + 2Re_L^0.5 cosh^-1 (y_max/D)) - ln⁡(1 + 2Re_L^0.5 cosh^-1 (y_min/D))]

where
Re = ρVD/μ is the Reynolds number
μ is the dynamic viscosity of the fluid
y_max is the maximum distance from the centerline of the pipe to the fluid velocity profile
y_min is the minimum distance from the centerline of the pipe to the fluid velocity profile

In this question, the velocity is zero over half of the cross-sectional area and is uniform over the remaining half. Therefore, y_max = D/2 and y_min = 0.

Substituting the values in the equation, we get:

f = (1/2) * [ln⁡(1 + 2Re_L^0.5 cosh^-1 (1)) - ln⁡(1 + 2Re_L^0.5 cosh^-1 (0))]

f = (1/2) * [ln⁡(1 + 2Re_L^0.5) - ln(1)]

f = (1/2) * ln⁡(1 + 2Re_L^0.5)

The momentum correction factor is given by:

C_f = (1 + f/2) / (1 - f/6)

Substituting the value of f, we get:

C_f = (1 + (1/2)ln⁡(1 + 2Re_L^0.5)) / (1 - (1/12)ln⁡(1 + 2Re_L^0.5))

Conclusion:
Therefore, the momentum correction factor for the given flow condition is 2 (option C).