Write short note on Power curve in mixing and agitation.

An empirical correlation of the power (or power number) with other variables of a system permits fairly accurate prediction of the requirement of a given impeller to rotate at a given speed. Such correlations can be obtained by using a method of dimensional analysis. The power requirement of the impeller is a functional of geometrical details of the impeller and vessel, the viscosity and the density of liquid, and the rotational speed of impeller. > > An empirical correlation that can be obtained for a given system from the dimensional analysis has the form as : >

${\P}/{\N^3 \D\a^5 \ρ} = \f ({\ND\a^2 \ρ}/{\μ} , {\N^2\D_\a}/{\ρ})$ —(1) > >

where ${\P}/{\N^3 \d\a^5 \ρ}$ is the power number $(\N\P)$, —(2)

${\ND\a^2 \ρ}/{\μ}$ is the impeller Reynold’s number $(\N{\Re})$, —(3)

${\N^2\D\a}/{\ρ}$ is the Froude number $(\N{\Fr})$, —(4)

N = Rotational speed in revolution per second, $\D_\a$ = Diameter of impeller, ρ = Density of fluid,
and μ = Viscosity of fluid. When $\N_{\Re} > 10,000$, the flow in the vessel is turbulent and when $\N_{\Re} < 10$, the flow is laminar For $\N_{\Re}$ between 10 and 10,000 the flow is in a transition region in which the flow is turbulent at the impeller and laminar in remote parts of the vessel. Equation (1) is also written as : $\N_\P = \f(\N_{\Re}, \N_{\Fr})$     ---(5) The Froude number, $\N_{\Fr}$represents the influence of gravitation and effects the power consumption only when vortex is present. If the speed of impeller is increased, in unbaffled vessels, the centrifugal force acting in the liquid causes the free surface of the liquid to assume a paraboloid form by raising the liquid level at the wall and lowering the level at the shaft. This is called _vortex._ The vortex is avoided by use of baffles. For Reynold's number < 300, vortex may not be observed even for the unbaffled vessel. The Reynold's number accounts for viscous forces and in agitated vessels, as usually the case, the viscous forces are significant. Thus, equation (5) reduces to $\N_\P = \f (\N_{\Re})$     ---(6)

The power consumption is related to the density and the viscosity of the liquid, the rotational speed, and the impeller diameter by plotting power as a functional of Reynold’s number.

At lower Reynold’s number (laminar flow), the relationship between $\N\P$ and $\N{\Re}$ may be given as

$\N\P = {\C0}/{\N_{\Re}}$ —(7)

where $\C_0$ is a constant for a given impeller and given geometrical details.

Rearranging equation (7), we get

$\N\P . \N{\Re} = \C_0$ —(8)

Substituting the values of $\N\p$ and $\N{\Re}$ in equation (8), we get

$\P = \C0 \μ \D\a^3 \N^2$ —(9)

Equation (9) indicates that if the speed is doubled, power consumption will increase by a factor of four.

For higher values of $\N_{\Re}$, the Froude number plays an important part. In this case, power number is constant i.e.

$\N_\p = \Constant = \C’$ —(10)

$\P = \C’ \ρ \D_\a^5 \N^3$ —(11)

Equation (11) indicates that if the speed is doubled, the power consumption increases by factor of eight in the turbulent flow region.

Curve 1 : Curve blade turbine, 4 baffles each width of baffle $\D\T ⁄ 12, \D\T -$tank diameter.

Curve 2 : Open straight blade (six blade) turbine four baffles each $\D_\T ⁄12.$

Curve 3 : Pitched blade turbine, 4 baffles each $\D_\T ⁄ 12$.

Curve 4 : Propeller, four baffles each 0.1 $\D\T$. Pitch = $2 \D\a$.

Curve 5 : Propeller, four baffles each 0.1 $\D\T$. pitch = $\D\a$.

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