Kinetic energy correction factor: Meaning, Formula, for laminar flow [with pdf]

For the flow of liquid, the rate of kinetic energy calculated by using average velocity and the actual kinetic energy rate have different values.

The kinetic energy correction factor gives the relation between kinetic energy by average velocity and the actual kinetic energy.

Therefore by using this factor, we can easily calculate the actual rate of kinetic energy.

What is Kinetic energy correction factor?

The ratio of kinetic energy based on the actual velocity of the fluid to the rate of kinetic energy based on average velocity is known as the kinetic energy correction factor.

The kinetic energy correction factor is denoted by the symbol ‘α’ and it is a unitless quantity.

From the statement, the formula of the kinetic energy correction factor is,

Kinetic energy correction factor

General equation for kinetic energy correction factor:

The kinetic energy correction factor is given by,

`\alpha= \frac{KE_{Actual\ velocity}}{KE_{AveraGe\ velocity}}`

At the first, we need to find the equation for the rate of kinetic energy for actual and average velocity.

I] Rate of kinetic energy based on actual velocity:

Consider an elementary portion with radius dr and area dA.

The mass flow rate through area dA is given by,

`d \dot{m}=\rho .dV`

`=\rho [dA.u]`

`=\rho [(2\pi r.dr).u]`

`d \dot{m}=2\rho \pi .u.r.dr`

Now, the rate of kinetic energy for actual velocity through the elemental surface is given by,

`d \dot{K}E_{actual}=\frac{1}{2}d\dot{m}.u^{2}`

`d \dot{K}E_{actual}=\frac{1}{2}[2\rho. \pi .u.r.dr]u^{2}`

`d \dot{K}E_{actual} =\rho. \pi. u^{3}.r.dr`

Now the total actual rate of kinetic energy for the circular cross-section of radius R is given by,

`\dot{KE}_{actual}=\int_{0}^{R}\rho. \pi. u^{3}.r.dr`

For the fluid with ρ = constant, the equation will become,

`\dot{KE}_{actual}=\rho. \pi\int_{0}^{R} u^{3}.r.dr`


II] Rate of kinetic energy for average velocity:

For the average velocity of V, the mass flow `\dot{m}` rate is given by,

`\dot{m}=\rho \times \dot{V}`

`\dot{m}=\rho [A\times V]`

`\dot{m}=\rho [\pi. R^{2}. V]`

Now the rate of kinetic energy for average velocity is given by,

`\dot{KE}_{averaGe}=\frac{1}{2}\dot{m}V^{2}`

`=\frac{1}{2}[\rho (\pi R^{2}V)]V^{2}`

`\dot{KE}_{averaGe}=\frac{1}{2}\rho. \pi. R^{2}.V^{3}`

Now by putting these values in the equation of kinetic energy correction factor,

`\alpha= \frac{KE_{Actual\ velocity}}{KE_{AveraGe\ velocity}}`

`=\frac{\rho \pi \int_{0}^{R}u^{3}.r.dr}{\frac{1}{2}\rho \pi R^{2}V^{3}}`

`\alpha =\frac{2 \int_{0}^{R}u^{3}.r.dr}{V^{3}R^{2}}` –[1]

This is the general equation for calculating the kinetic energy correction factor.

Kinetic energy correction factor for laminar flow:

For laminar flow,

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

Average velocity is given by,

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

Put the values of u and V in the above equation [1] of α,

`\alpha` =`\frac{2\int_{0}^{R}[\frac{1}{4\mu }(-\frac{dP}{dx})(R^{2}-r^{2})]^{3}r.dr}{[\frac{1}{8\mu }(-\frac{dp}{dx})R^{2}]^{3}R^{2}}`

`=\frac{16\int_{0}^{R}(R^{2}-r^{2})^{3}r.dr}{R^{8}}`

=`\frac{16\int_{0}^{R}(R^{6}-r^{6}-3R^{4}r^{2}+3R^{2}r^{4})r.dr}{R^{8}}`

=`\frac{16\int_{0}^{R}(R^{6}r-r^{7}-3R^{4}r^{3}+3R^{2}r^{5})dr}{R^{8}}`

=`\frac{16}{R^{8}}`x`[R^{6}\frac{r^{2}}{2}-\frac{r^{8}}{8}-3\frac{R^{4}r^{4}}{4}+3\frac{R^{2}r^{6}}{6}]_{0}^{R}`

=`\frac{16}{R^{8}}[\frac{R^{8}}{2}-\frac{R^{8}}{8}-3\frac{R^{8}}{4}+3\frac{R^{8}}{6}]`

α =`\frac{16}{R^{8}}(\frac{R^{8}}{8})`

α = 2

Therefore for laminar flow, the value of the kinetic energy correction factor is 2.

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