6.3 Power Required for Agitation

\[\text{Po} = \phi(\text{Re}, \text{Fr}, W/D, H/D, \text{etc.})\] where

\(\text{Po}\) = \(\displaystyle \dfrac{P}{n^3D_a^5\rho}\) = Power number
\(\text{Re}\) = \(\displaystyle \dfrac{nD_a^2\rho}{\mu}\) = Impeller Reynolds number
\(\text{Fr}\) = \(\displaystyle \dfrac{n^2D_a}{g}\) = Froude number
\(P\) = power required for agitation
\(n\) = agitator rpm
\(D_a\) = diameter of impeller
\(\rho\) = density of fluid
\(\mu\) = viscosity of fluid
\(g\) = acceleration due to gravity

  • The Froude number is usually important only in situations where gross vortexing exists and this can be neglected if the Reynolds number is less than about 300. For higher Reynolds numbers the Froude number effects are eliminated by the use of baffles or off-centre stirring. Thus in cases where the Froude number can be neglected we have: \[\text{Po} =\phi(\text{Re}, \text{geometrical ratios})\] and if we consider geometrically similar systems, \[\text{Po} = \phi(\text{Re})\]

  • The Power number versus Reynolds number curve has similarity to the friction factor vs. Reynolds number behavior for pipe flow. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant.

  • The shape of the tank has relatively little effect on \(\text{Po}\). Circulation patterns, of course, are strongly affected by tank shape.

  • Froude Number: Froude number (\(\text{Fr}\)) is a measure of the ratio of the inertial stress to the gravitational force per unit area acting on the fluid. It appears in fluid-dynamic situations where there is significant wave motion on a liquid surface. It is significantly important in ship design. In unbaffled tanks, a vortex forms and the Froude number has an effect. Froude number is usually important only in situations where gross vortexing occurs, and can be neglected if the value of Reynolds number is less than about 300. For higher values of \(\text{Re}\), the Froude number seems to exert some influence on the value of \(\text{Po}\). However, this effect may be minimized, or indeed eliminated, by the use of baffles or installing the impeller off-centre.

  • For low \(\text{Re}\), (\(\text{Re} < 10\)), \(\text{Po}\) versus \(\text{Re}\) is a straight line with a slope of \(-1\) on logarithmic coordinates, for both baffled and unbaffled tanks. \[\text{Po} \cdot \text{Re} = \frac{P}{n^2D_a^3\mu} = K_L \quad \text{ (a constant)}\] from which \[P = K_Ln^2D_a^3\mu\] For \(\text{Re}\) less than 10, the flow is laminar and density is no longer a factor and power delivered to the liquid is proportional to \(D_a^3\) for both baffled and unbaffled tanks.

  • In baffled tanks at Reynolds number larger than about 10,000 the power number is independent of the Reynolds number, and viscosity is not a factor. \[\text{Po} = K_T \quad \text{ (a constant)}\] from which \[P = K_Tn^3D_a^5\rho\]

  • Weber Number \[\text{We} = \frac{\rho_cn^2D_a^3}{\sigma} = \frac{\text{kinetic energy}}{\text{surface tension stress}}\] where

    \(\rho_c\) = density of continuous liquid phase
    \(\sigma\) = interfacial tension

    For two-phase dispersions Weber number is a significant factor.