Momentum correction factor

The momentum correction factor makes it easier to calculate the rate of momentum based on the actual velocity. As a result, rather than using actual flow velocity, we can utilize average velocity to find the rate of momentum based on actual velocity.

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What is momentum correction factor?

The momentum correction factor is the ratio of momentum rate based on the actual velocity to the momentum rate based on average velocity. It shows how momentum rate based on actual velocity and the momentum rate based on average velocity are related.

It is a unitless quantity and is denoted by the symbol ‘β’.

The value of the momentum rate based on the actual velocity (\dot{P}_{\text{Actual velocity}}) is different from the momentum rate based on the average velocity. Thus the momentum correction factor helps to find the relation between the momentum rate based on actual velocity and the momentum rate based on average velocity.

Mathematically it is expressed as,

Momentum correction factor, \beta = \frac{\dot{P}_{\text{Actual velocity}}}{\dot{P}_{\text{Average velocity}}}

Momentum correction factor equation:

The momentum correction factor is given by,

Where,
\dot{P}_{\text{Actual velocity}} = Momentum rate based on the actual velocity
\dot{P}_{\text{Average velocity}} = Momentum rate based on the average velocity

General equation for momentum correction factor:

A per the definition, the momentum correction factor is given by,

β = \frac{\dot{P}_{\text{Actual velocity}}}{\dot{P}_{\text{Average velocity}}} —-Equation[1]

To find the general equation of the β for the circular pipe, it is necessary to find the momentum rate based on actual velocity and the momentum rate based on average velocity.

1] Momentum rate based on actual velocity:

For the fluid flowing through the circular pipe of radius ‘R’, consider an elemental area of thickness dr located at a distance ‘r’ from the center.

The area of the smaller portion is given by,

dA = 2 π r.dr

The mass flow rate through this smaller portion can be given by,

d\dot{m} = ρ dA.u

Where u is the velocity of flow at radius r and ρ is the density of the fluid.

By putting the value of dA in the equation of d\dot{m},

d\dot{m} = ρ.(2.π.r.dr).u

d\dot{m} = 2.ρ.π.r.dr.u

The rate of momentum for the fluid flowing through smaller portion is given by,

d\dot{P}_{\text{actual velocity}} = d\dot{m} x u

d\dot{P}_{\text{actual velocity}} = {2ρπr.dr.u} x u

d\dot{P}_{\text{actual velocity}} = 2 ρ π r.dr. u²

To find the total actual momentum rate, integrate the both sides of the equation.

\int d\dot{P}_{\text{actual velocity}} = \int_{0}^{R}2 ρ π r.dr. u²

\mathbf{\dot{P}_{\text{actual velocity}}} = 2ρπ \mathbf{\int_{0}^{R}}r. u² .dr —-Equation[2]

2] Momentum rate based on the average velocity:-

For the same cross-section and for the fluid flowing with an average velocity of ‘V’, the mass flow rate is given by,

\dot{m} = ρ A V

\dot{m} = ρ (πR²) V ——-[∵ A = πR²]

Now the momentum rate based on the average velocity is given by,

\dot{P}_{\text{average velocity}} = \dot{m}V

\dot{P}_{\text{average velocity}} = ρ (πR²) V . V

\mathbf{\dot{P}_{\text{average velocity}}} = ρ π R² V² — Equation [3]

Now put the values of \dot{P}_{\text{actual velocity}} and \dot{P}_{\text{average velocity}} in the equation [1] of β.

β = \frac{\dot{P}_{\text{actual velocity}}}{\dot{P}_{\text{average velocity}}} = \frac{2\pi \rho \int_{0}^{R}r.u^{2}.dr}{\rho \pi R^{2} V^{2}}

This is the general equation of momentum correction factor for flow through the circular cross-section.

Momentum correction factor for laminar flow:

For the steady and fully developed laminar flow through the pipe, the equation of velocity is given by,

u = \frac{1}{4\mu }(\frac{-dp}{dx})(R^{2}-r^{2})

And the average velocity is given by,

v = \frac{1}{8\mu }(\frac{-dp}{dx})R^{2}

Put these values into the equation of β

β = \frac{2 \int_{0}^{R}r.u^{2}.dr}{R^{2} V^{2}}

β = \frac{2\int_{0}^{R}r[\frac{1}{4\mu }(-\frac{dp}{dx})(R^{2}-r^{2})]^{2}.dr}{R^{2}[\frac{1}{8\mu }(\frac{-dp}{dx})R^{2}]^{2}}

β = \frac{8\int_{0}^{R}r[R^{2}-r^{2}]^{2}.dr}{R^{6}}

β = \frac{8\int_{0}^{R}r[R^{4}-2R^{2}r^{2}+r^{4}].dr}{R^{6}}

β = \frac{8\int_{0}^{R}[R^{4}r-2R^{2}r^{3}+r^{5}].dr}{R^{6}}

β = \frac{8[\frac{R^{6}}{2}-\frac{2R^{6}}{4}+\frac{R^{6}}{6}]}{R^{6}}

Thus for the steady and fully developed laminar flow through the circular pipe, the value of the momentum correction factor is 1.33.

Significance of momentum correction factor:

The momentum correction factor gives the relation between the momentum rate based on actual velocity and the momentum rate based on the average velocity.

Thus it helps to find the actual momentum rate by using the average velocity.

Momentum correction factor examples:

Given:
V = 5 m/s
R = 25 mm = 0.025 m
β = 1.33
\rho_{water} = 1000 Kg/m³

Solution:

The mass flow rate is given by,

\dot{m} = ρ.A.V

\dot{m} = ρ [π.R²] V

\dot{m} = 1000 [π x 0.025²] x 5

\mathbf{\dot{m}} = 9.817 Kg/s

The mometum rate based on average velocity is given by,

\dot{P}_{\text{Average velocity}} = \dot{m} \times V

\dot{P}_{\text{Average velocity}} = 9.817 x 5

\mathbf{\dot{P}_{\text{Average velocity}}} = 49.085 Kg.m/s²

The actual rate of momentum is given by,

β = \frac{\dot{P}_{\text{Actual velocity}}}{\dot{P}_{\text{Average velocity}}}

1.33 = \frac{\dot{P}_{\text{Actual velocity}}}{49.085}

This is the actual rate of momentum for the flow.

FAQs:

1. What is the momentum correction factor for laminar flow in a circular tube?

For the steady and fully developed laminar flow in a circular pipe, the value of the momentum correction factor is 1.33.

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